Chapter 0 – Solutions
Circles
Learning Goal:
To calculate the circumference or area of a circle.
Every day, we see circles in compact disks, coins, and wheels, just to name a few examples.
Circles are also very common in math and physics, because they are easy to describe and
calculate with. There is a very simple definition of a circle: Pick a point A and a distance r.
A circle with center A and radius r is the set
of all points a distance r from the point A. For instance, if you pick the center of your screen as A
and 3 cm s r, then all of the points on the screen that are 3 cm away from the center form a circle.
An easy way to draw a circle is to hold a string fixed at one end (for instance, with a pin) and
attach a pen to the other end. Holding the string taut and drawing wherever the taut string allows
gives a circle. This is the same technique you follow when using a compass. With a compass, the
string is simply replaced by a metal or plastic structure, which usually has some markings to let
you pick the radius of your circle.
The radius of a circle is the only measure that you need to determine any other measure, such as
the circumference or area. For instance, the circumference C of a circle (the length measured
around the outside of the circle, i.e., the perimeter) is C=2πr.
Suppose that you have a piece of string 7 cm long. If one end is held fixed and you draw with a
pen at the other end, keeping the string tight, then you will draw a circle.
Part A
What is the circumference of this circle?
Express your answer in centimeters to three significant figures.
ANSWER:
C= 44.0 cm
Suppose that you needed to make a pen for some small animals. You have 12 m of fencing. You
decide to make a circular pen, because if you wish to enclose an area using a given length of
fencing, then a circular fence encloses a larger area than a fence of any other shape.
Part B
What is the radius r of the pen?
Express your answer in meters to four significant figures.
ANSWER:
r= 1.910 m
Part C
What is the area A of the circular enclosure from Part B?
Express your answer in square meters to three significant figures.
ANSWER:
A= 11.5 m2
In Part B, it was stated that a circular enclosure gives the greatest area for a given perimeter. If
you wrote the areas of an equilateral triangle, a square, and a circle for a given perimeter P, you
would find
,
,
and
.
The formula for the circle has the smallest denominator and thus the largest area for the given
perimeter. This is true regardless of what shapes you choose to compare to the circle.
Part D
If you swim from one point at the edge of the pool to another, along a straight line, what is the
longest distance d you can swim?
Express your answer in meters to three significant figures.
Hint 1. How to approach the problem
The longest distance between two points on a circle is the diameter of the circle. The diameter is
a line through the center of the circle. Think about how the length of such a line relates to the
radius. Drawing a picture may help if you are having trouble seeing how the two relate.
You have the area of the pool and would like to know the radius. Since the formula A=πr2 relates
the area to the radius, you can use this formula to find the radius, using your given value of the
area.
Hint 2. A helpful picture
The picture below shows a diameter of a circle, which is the longest segment that can be drawn
connecting two points on a circle. Notice that since the diameter goes through the center of the
circle, you can break it up into two radii.
ANSWER:
d= 5.64 m
Converting Units
The ability to convert from one system of units to another is important in physics. It is often
impractical to measure quantities in the standard meters, kilograms, and seconds, but the laws of
physics that you learn will involve constants that are defined in these units. Therefore, you may
often have to convert your measured quantities into meters, kilograms, and seconds.
The following table lists metric prefixes that come up frequently in physics. Learning these
prefixes will help you in the various exercises.
mega- (
)
kilo- ( )
centi- ( )
milli- ( )
micro- ( )
nano- ( )
When doing unit conversions, you need a relation between the two units. For instance, in
converting from millimeters to meters, you need to know that
.
Once you know this, you need to divide one side by the other to obtain a ratio of m to mm
.
If you are converting from millimeters to meters, then this is the proper ratio. It has mm in the
denominator, so that it will cancel the units of the quantity that you are converting. For instance,
if you were converting 63 mm then you would have
.
If you were converting a quantity from meters to millimeters, you would use the reciprocal ratio:
.
Part A
Suppose that you measure a pen to be 10.5 cm long. Convert this to meters.
Express your answer in meters.
Hint 1. Relating centimeters and meters
To solve this problem, you will need to use the relation 100 cm = 1 m. You can determine such
relations using the metric prefixes given in the introduction to this problem. If one centimeter
equals 10-2 meters, then you need 102 centimeters to equal a whole meter, just as you know that
if one quarter equals 4-1 US dollars, then you need 41 quarters to equal a whole US dollar.
ANSWER:
10.5 cm = 0.105 m
Part B
Suppose that, from measurements in a microscope, you determine that a certain bacterium covers
an area of 1.50 µm2. Convert this to square meters.
Express your answer in square meters.
Hint 1. Find the conversion factor
Which of the following gives the proper conversion factor to use? From the table in the
introduction, you can see that 1 µm = 10-6 m, which gives 106 µm = 1 m.
ANSWER:
1.50 µm2= 1.50×10−12 m2
Part C
Suppose that you find the volume of all the oceans to be 1.4 X 109 m3 in a reference book. To
find the mass, you can use the density of water, also found in this reference book, but first you
must convert the volume to cubic meters. What is this volume in cubic meters?
Express your answer in cubic meters.
Hint 1. Find the conversion factor
Which of the following gives the proper conversion factor to use? From the table in the
introduction, you can see that 1 km = 103 m.
ANSWER:
1.4 x109 km3 =
1.40×1018 m3
Part D
In a laboratory, you determine that the density of a certain solid is 5.23x10-6 kg/mm3. Convert
this density into kilograms per cubic meter.
Notice that the units you are trying to eliminate are now in the denominator. The same principle
from the previous parts applies: Pick the conversion factor so that the units cancel. The only
change is that now the units you wish to cancel must appear in the numerator of the conversion
factor.
Express your answer in kilograms per cubic meter.
Hint 1. Find the conversion factor
Which of the following gives the proper conversion factor to use? From the table in the
introduction, you can see that 1 mm = 10-3 m. Recall that you are trying to cancel units out of the
denominator of the fraction.
ANSWER:
5230
=
You are now ready to do any sort of unit conversion. You may encounter problems that look far
more complex than those you've done in this problem, but if you carefully set up conversion
factors one at a time to cancel the units you don't want and replace them with the units that you
do want, then you will have no trouble.
Dealing with Numbers from Tiny to Huge
Learning Goal:
To understand powers of 10 and scientific notation.
Physics deals with an incredible range of objects and phenomena: from processes involving
subatomic particles to those involving the entire universe. This variety leads to sizes from the
width of an atom to the distances between stars. Physicists use several methods to make dealing
with very large or very small numbers more convenient.
We will start exploring these methods by considering two numbers: the lifetime of a particle
called a top quark and the age of the universe.
The age of the universe is around 100,000,000,000,000,000 s. A top quark has a lifetime of
roughly 0.000000000000000000000001 s. Writing numbers out with all these zeros is not very
convenient. Such quantities are usually written as powers of 10. The age of the universe can be
written as 1017 s and the lifetime of a top quark as 10-24 s.
Note that 1017 means the number you would get by multiplying 10 times 10 times 10... a total of
17 times. This number, as you can see above, would be a one followed by seventeen zeros.
Similarly, 10-24 is the result of multiplying 0.1 (or 1/10) times itself 24 times. As seen above, this
is written as 23 zeros after the decimal point followed by a one.
Part A
How many top quark lifetimes have there been in the history of the universe (i.e., what is the age
of the universe divided by the lifetime of a top quark)? Note that these powers of 10 follow the
same rules that any exponents would follow.
Express your answer numerically.
Hint 1. Multiplying and dividing with exponents
Recall that when you multiply two numbers with the same base (e.g., 24 and 27), you obtain the
result by adding the exponents (e.g., 24 * 27 = 24+7 = 211). When you divide, you subtract the
exponents. For example,
.
ANSWER:
1.00×1041
Part B
Compute (4.29x1015)*(1.51x10-4).
Express your answer to three significant figures.
ANSWER:
6.48×1011
If you had been asked to multiply 4.29x1015 times 3.00x10-4 , you would have found (to three
significant figures) 12.9x1011. You should rewrite this in the more accepted form 1.29x1012 .
(Notice that you must increase the exponent to balance moving the decimal point to the left.)
Part C
Compute 6.28x1013 + 8.30x1011.
Express your answer to three significant figures
Hint 1. How to approach the problem
Of the two powers of 10, 11 is the smallest. Convert 8.30x1011 to some number times 1013, then
add as usual.
ANSWER:
6.36×1013
Part D
Suppose that a certain fortunate person has a net worth of $76.0 billion ($7.60x1010). If his stock
has a good year and gains $3.20 billion (3.20x109) in value, what is his new net worth?
Express your answer to three significant figures.
Hint 1. How to approach the problem
Of the two powers of 10, 9 is the smallest. Convert 3.20x109 to some number times 1010, then
add as usual.
ANSWER:
7.92×1010 dollars
Part E
Suppose that this individual now decides to give one-eighth of a percent of his new net worth to
charity. How many dollars are given to charity?
Express your answer to three significant figures.
Hint 1. Percent and scientific notation
Recall that 50 percent is the same as 50/100 or 50 x 10-2. Knowing that
you can rewrite "one-eighth of a percent" as 0.125x10-2 = 1.25x10-3.
ANSWER:
9.90×107 dollars
Interpreting Graphs
Learning Goal:
To be able to gain many different types of information from a graph.
Being able to read graphs is an important skill in physics. It is also critical in day-to-day life, as
information in the news and in business meetings is often presented in graphical form. In this
problem, you will consider a single graph
and
all of the information that can be gained from it. Since the graph axes have no labels, think of it
as a graph of something important to you, whether that is GPA, your bank balance, or something
else. Specific applications will be noted for each way of analyzing a graph.
The easiest information to obtain from a graph is its value at a point. The height of the graph
above the horizontal axis gives the value of the graph. Points above the horizontal axis have
positive values, whereas points below the axis have negative values. The vertical axis will
usually have specific values marked off so that you can tell exactly what value each height
corresponds to. In the graph you've been given, there are no exact values labeled, but you can
still tell relative values; you can make statements such as, "At point D, the graph has a greater
value than at point C."
Part A
At which point(s) does the graph have a positive value?
Enter all of the correct letters in alphabetical order. For instance, if you think that the correct
choices are B and F, you would enter BF.
Hint 1. Determining positive values from a graph
Any value above the horizontal axis is positive; any below is negative. Where the graph
intersects the horizontal axis, the value of the graph is zero, which is neither positive nor
negative.
ANSWER:
ABCDEF
The difference between positive and negative is important in many situations, for instance on
your bank statement. In physics it makes a big difference in many scenarios. Positive position
means to the right or above some reference point; negative position means to the left or below
the reference point. Positive velocity means moving to the right, whereas negative velocity
means moving to the left.
The graph is often more convenient than a table of numbers or an equation, because you can
immediately see where the graph takes positive values and where it takes negative values. With
an equation or a table of numbers, this would take some algebra or guess work.
Part B
At which point does the graph have its maximum value?
Enter the correct letter.
ANSWER:
E
Part C
Look at the graph from the introduction. The three points C, D, and F are all on straight segments.
Rank them from greatest rate of change to least rate of change.
Hint 1. Slope at point D
When the graph is horizontal, the slope of the graph is zero. You can see this by noticing that the
change in vertical position as you move from left to right across a horizontal graph is zero. No
matter what the change in horizontal position is, when you divide to find the slope it will be zero.
ANSWER:
Part D
At which point is the graph increasing at the greatest rate? For now, ignore point E. We will
discuss it after this part.
Enter the correct letter.
Hint 1. Drawing the tangent
To find the tangent line at a particular point, you should draw a dot at that point on the graph and
then draw pairs of points, one on either side of the point you care about, that are the same
distance from the point you care about.
If you then connect those pairs of dots, the lines connecting them will get closer and closer to the
proper slope as you move to pairs that are closer and closer to the point you care about. Once you
get pretty close to that point, you should be pretty confident of the slope for the tangent line.
With practice, you will gain an intuitive ability to see roughly how the slope of the tangent at a
point should look.
ANSWER:
D
You were told to ignore point E for this part. This is because the rate of change is not well
defined at sharp corners. You won't ever be asked for the rate of change of a graph at a sharp
corner, though points near the corner should have well-defined rates of change.
Points B and C are also special, because the slope at those points is zero. This should be easy to
see at C, since the graph is actually a horizontal line in the area near C. If you carefully work out
the tangent at point B using the method described in the hint for this part, you will see that the
tangent is horizontal there as well. Since a horizontal tangent has a slope of zero, which is neither
negative nor positive, the graph is neither increasing nor decreasing at points B and C.
Part E
At which point(s) is the graph decreasing?
Enter all of the correct letters in alphabetical order. For instance, if you think that the correct
choices are B and F, you would enter BF.
ANSWER:
FGH
Part F
You wish to find the area under the graph between the origin and some point on the graph.
Which point will yield the greatest area?
Enter the correct letter.
ANSWER:
G
Part G
You are looking at the area under the graph between two points. The area is zero. Which two
points are you looking at?
Enter the two letters in alphabetical order. For instance, if you think that the correct choice is B
and F, you would enter BF.
Hint 1. How to approach the problem
Since you want the area under the curve between two points to be zero, the graph must define
equal-sized shaded regions above and below the horizontal axis. Look for two points that are
near each other, one above and one below the horizontal axis.
ANSWER:
FH
Multiplying and Dividing Fractions
Learning Goal:
To understand the multiplication and division of fractions.
If you had eight quarters, you could likely figure out relatively quickly that this amounted to two
dollars. Although this may be purely intuitive, the underlying math involves multiplication of
fractions. The value of eight quarters is the same as
.
When you multiply fractions, you multiply the numerators (the top numbers in the fractions) to
get the numerator of the answer, and then multiply the denominators (the bottom numbers in the
fractions) to find the denominator of the answer. In this example,
,
giving 2, as you expected. Similarly, asking for a fraction of a fraction (e.g., "one fifth of a
quarter") is a case of multiplying fractions:
.
In this problem, before entering your answer, be sure to reduce your fraction completely. If you
get 8/ 6 for your answer, reduce it to 4/3 before entering it, or else it will be marked wrong. Also,
don't worry if the numerator is larger than the denominator. It is almost always easier and more
useful to further calculations to leave such answers as improper fractions rather than to convert
them to mixed numbers such as 1&1/3.
Part A
If you have a quarter of a pie and you cut it in half, what fraction of a pie would each slice
represent?
Give the numerator followed by the denominator, separated by a comma.
Hint 1. Setting up the equation
The problem is asking you for the value of "half of a quarter." This translates into math as
.
Multiply the numerators to find the numerator of the answer. Then, multiply the denominators to
find the denominator of the answer.
ANSWER:
1, 8
Part B
Find the value of
.
Though these numbers aren't quite as nice as the ones from the example or the previous part, the
procedure is the same, so this is really no more difficult.
Give the numerator followed by the denominator, separated by a comma.
Hint 1. Find the numerator
To find the numerator, simply multiply the numerators of the two fractions (17 and 2). What is
the result?
ANSWER:
34
Hint 2. Find the denominator
To find the denominator, simply multiply the denominators of the two fractions (15 and 11).
What is the result?
ANSWER:
165
ANSWER:
34, 165
Dividing fractions is no more difficult than multiplying them. Consider the problem
.
Notice that division by 3 is identical to multiplication by 1/3, because both operations consist of
breaking the first number into three parts. Thus
.
The only new step in division of fractions is that you must invert (flip) the second fraction. Then,
simply multiply as shown here.
Part C
Consider the following equivalent expressions:
and
.
What are the values of a and b?
Give the value of a followed by the value of b, separated by a comma.
ANSWER:
12, 13
Part D
Calculate the value of
.
Give the numerator followed by the denominator, separated by a comma.
ANSWER:
9, 26
Part E
Now, find the value of
.
Don't be intimidated by the complexity of this expression. Finding this value consists of simply
multiplying twice and then dividing once, tasks that are no more difficult than what you've done
before.
Give the numerator followed by the denominator, separated by a comma.
Hint 1. How to approach the problem
Break the problem down into individual operations. Notice that
is the same as
.
This second form makes the individual operations more obvious.
First, multiply
.
Then, multiply
.
Finally, divide the two fractions that you have found.
Hint 2. Find the value of 3/16 * 2/5
What is the value of 3/16 * 2/5?
Give the numerator followed by the denominator, separated by a comma.
ANSWER:
3, 40
Hint 3. Find the value of 7/4 * 3/2
What is the value of 7/4 * 3/2?
Give the numerator followed by the denominator, separated by a comma.
ANSWER:
21, 8
ANSWER:
1, 35
Special Triangles
Learning Goal:
To understand the properties of special types of triangle.
There are several special types of triangle that you will encounter often in physics.
Understanding the properties of special triangles is particularly useful in working with vectors
but may also arise in other contexts such as optics.
Part A
An equilateral triangle is a triangle with all three sides of equal length. All of the angles in an
equilateral triangle are equal.
What is the measure of angle θ in the triangle shown?
Recall that the sum of the angles in a triangle
equals 180 .
Express your answer in degrees.
ANSWER:
60
Since the sum of the angles in any triangle is 180 , all equilateral triangles will have three 60
angles.
Part B
Dividing an equilateral triangle in half leads to the next special triangle that we will look at.
Consider the figure
showing an equilateral
triangle with a line dividing it down the middle. This line divides both the angle at the top of the
triangle and the base of the triangle into two equal parts.
Find the measure of angle ϕ in degrees and the length of segment x in centimeters.
Express the two answers in degrees and centimeters, respectively. Separate the two with a
comma.
ANSWER:
ϕ, x= 30, 7 degrees, cm
Dividing an equilateral triangle in half, as we just did here, yields a triangle often called a 30-60
right triangle, in reference to the measures of its acute angles in degrees.
Part C
What is the length y of the remaining, vertical side of the 30-60 right triangle?
Express your answer in centimeters to three significant figures.
Hint 1. Use the Pythagorean Theorem
You can find the value of y using the Pythagorean Theorem. The Pythagorean Theorem says that
a2 + b2 = c2, where a and b are the lengths of the legs of a right triangle and c is the length of the
hypotenuse (the side opposite the right angle). Which of the following gives the proper values of
a, b, and c for this triangle?
ANSWER:
ANSWER:
= 12.1
In a right triangle with acute angles of measure 30 and 60 , the sides will always be in the ratio
1:
: 2, for the sides opposite the 30 , 60 , and 90 angles, respectively.
Part D
In a 45-45 right triangle, the two legs have the same length. In the figure
both are given lengths of 1
length of the hypotenuse of this triangle?
. What is the
Express your answer in centimeters to four significant figures.
Hint 1. Use the Pythagorean Theorem
You can find the value of x using the Pythagorean Theorem. The Pythagorean Theorem says that
a2 + b2 = c2, where a and b are the lengths of the legs of a right triangle and c is the length of the
hypotenuse (the side opposite the right angle). Which of the following gives the proper values of
a, b, and c for this triangle?
ANSWER:
ANSWER:
1.414
In a right triangle with two acute angles of measure 45 , the sides will always be in the ratio 1: 1:
, for the sides opposite the 45 , 45 , and 90 angles, respectively.
Pythagorean Theorem
Learning Goal:
To understand and apply the Pythagorean Theorem.
The Pythagorean Theorem is named after a religious school from ancient Greece whose students
believed whole numbers to be the foundations of the universe. They discovered much interesting
math using whole numbers. However, the discovery that they are most famous for also led to the
downfall of their religion! The Pythagorean Theorem leads directly to the discovery of irrational
numbers—numbers that cannot be written as the ratio of two whole numbers. Seeing that even
something as simple as the diagonal of a square leads to irrational numbers shattered their belief
in the holiness of whole numbers, but this insight also laid the foundation for many of the
discoveries that made Greek mathematics, particularly geometry, so successful.
The Pythagorean Theorem relates the lengths of the two legs (the sides opposite the two acute
angles) a and b of a right triangle to the length of the hypotenuse (the side opposite the right
angle) c. Given a right triangle as shown in the figure
, the Pythagorean Theorem is written
.
For instance, if you had a right triangle with legs both of length 1 (i.e., a=b=1), then the
Pythagorean Theorem would give
,
so that c=
.
Part A
Now, consider a right triangle with legs of lengths 5 cm and 12 cm . What is the length c of the
hypotenuse of this triangle?
Express your answer to three significant figures.
ANSWER:
c= 13 cm
Part B
Suppose that you have measured a length of 6 cm on one board and 8 cm on the other. You
would adjust the two boards until the length of the string had value c to ensure that the boards
made a right angle. What is c?
Express your answer in centimeters to three significant figures.
ANSWER:
c= 10 cm
A set of three integers that form the legs and hypotenuse of a right triangle is called a
Pythagorean triple. Any Pythagorean triple multiplied by an integer is another Pythagorean
triple. For instance, since this problem showed you that 6, 8, 10 is a Pythagorean triple, you also
know that 12, 16, 20 is a Pythagorean triple (found by doubling 6, 8, and 10).
Part C
Use the Pythagorean Theorem to determine which of the following give the measures of the legs
and hypotenuse of a right triangle.
Check all that apply.
ANSWER:
3, 4, 5
4, 11, 14
9, 14, 17
8, 14, 16
8, 15, 17
Part D
What is the length x of a side of the small inner square?
Express your answer in terms of the variables a and b.
Hint 1. A more helpful figure
In the figure below, the side of length a for the yellow triangle plus the side of the small square
makes up the side of length b for the light blue triangle. You can set up a simple equation
relating a, b, and x using this fact. Solve this equation for x.
ANSWER:
x= b -a
Part E
Given that the side of the square has a length b - a, find the area of one of the four triangles and
the area of the small inner square.
Give the area of one of the triangles followed by the area of the small inner square separated by a
comma. Express your answers in terms of the variables a and b.
Hint 1. Finding the areas
Recall that the formula for the area of a triangle is A = (1/2)bh, where b is the length of the base
and h is the height. For a right triangle, the two legs form the base and height.
Also, recall that the formula for the area of a square is A = s2, where s is the length of a side of
the square.
ANSWER:
=
,
Since the entire square is made up of four such triangles plus the small square, the area of the
entire square is
Since the entire square has sides of length c , you could also simply write
.
Equating the two expressions for the area of the entire square gives
,
the Pythagorean Theorem!
The Root Is the Problem
Learning Goal:
To understand and be able to perform arithmetic involving square roots.
Many times in physics, you will deal with quantities involving square roots. It is important that
you be able to easily handle such quantities, so that difficulties in arithmetic don't hinder your
understanding of the physics that leads to them.
The standard form for a number involving a square root is a , where b is a number that cannot
be divided by a perfect square. The formula a means "a multiplied by the square root of b."
Don't think of 3 as being any more complicated than 3*2, because both are just simple
multiplication.
Given an expression such as
, where the number under the square root sign is divisible by a
perfect square, you can transform it into the standard form. To do this you will need to make use
of an important property of square roots:
.
This says that you can break up the number under the square root into the perfect square and the
leftovers. For example (using the fact that 4= 2*2),
.
Part A
Express
in the form a .
Give the value of a followed by the value of b separated by a comma.
Hint 1. Factors of 75
The prime factorization of 75 is
.
ANSWER:
5, 3
Part B
Find the product of
and 6
. Express it in standard form (i.e., a ).
Give the value of a followed by the value of b separated by a comma.
Hint 1. Multiply the two numbers before reducing
Think of the two numbers as 1
and 6
, so that you can see both coefficients and both
radicals explicitly. Find the product of the coefficients (1 and 6) and the product of the numbers
under the square roots (30 and 10).
Enter the product of the coefficients followed by the product of the numbers under the square
roots separated by a comma.
ANSWER:
6, 300
Now you know that
Look at the factors of 300 to determine whether it is
divisible by a perfect square.
ANSWER:
60, 3
Part C
Which of the following can be reduced to a single number in standard form?
ANSWER:
Part D
Find
in standard form.
Give the value of a followed by the value of b separated by a comma.
Hint 1. Express
in standard form
The prime factorization of 126 is
.
Using this, express 126 in standard form.
Give the value of a followed by the value of b separated by a comma.
ANSWER:
3, 14
Hint 2. Express
in standard form
The prime factorization of 56 is
.
Using this, express 56 in standard form.
Give the value of a followed by the value of b separated by a comma.
ANSWER:
2, 14
ANSWER:
5, 14
Proportional Reasoning
Learning Goal:
To understand proportional reasoning for solving and checking problems.
Proportional reasoning involves the ability to understand and compare ratios and to produce
equivalent ratios. It is is a very powerful tool in physics and can be used for solving many
problems. It's also an excellent way to check answers to most problems you'll encounter.
Proportional reasoning is something you may already do instinctively without realizing it.
Part A
You are asked to bake muffins for a breakfast meeting. Just as you are about to start making
them, you get a call saying that the number of people coming to the meeting has doubled! Your
original recipe called for three eggs.
How many eggs do you need to make twice as many muffins?
Express your answer as an integer.
ANSWER:
6
Part B
You have a dozen eggs at home, and you know that with them you can make 100 muffins. If you
find that half of the eggs have gone bad and can't be used, how many muffins can you make?
Express your answer as an integer.
ANSWER:
50
Recall that dividing a variable is the same as multiplying it by a fraction. If you keep this in mind,
then you can change this problem from "the number of eggs are divided by two" into "the
number of eggs are multiplied by one-half," which works just as any other multiplication. If you
look at the graph for the linear relationship, dividing by 2 is like moving from the middle point to
the left point marked on the graph.
Part C
When sizes of pizzas are quoted in inches, the number quoted is the diameter of the pizza. A
restaurant advertises an 8-inch "personal pizza." If this 8-inch pizza is the right size for one
person, how many people can be fed by a large 16-inch pizza?
Express your answer numerically.
Hint 1. How to approach the problem
The area of a pizza is what determines how many people can be fed by the pizza. You know that
the area of a circle is proportional to the square of the radius. Since the radius is proportional to
the diameter, it follows that the area is also proportional to the square of the diameter. Use this
relation to determine how the area, and therefore the number of people fed, changes.
ANSWER:
4
Part D
If a car is speeding down a road at 40 miles/hour (mph), how long is the stopping distance D40
compared to the stopping distance D25 if the driver were going at the posted speed limit of 25
?
Express your answer as a multiple of the stopping distance at 25 mph . Note that D25 is already
written for you, so just enter the number.
Hint 1. Setting up the ratio
Since 40/25 = 1.6, the car is moving at a speed 1.6 times the speed limit of 25 mph. The stopping
distance is proportional to the square of the initial speed, so the stopping distance will increase
by a factor of the square of 1.6.
ANSWER:
= 2.56
The quadratic relationship between stopping distance and initial speed is part of the reason that
speeding fines are doubled in school zones: At low speeds, a small change in speed can lead to a
large change in how far your car travels before it stops.
Part E
A construction team gives an estimate of three months to repave a large stretch of a very busy
road. The government responds that it's too much inconvenience to have this busy road
obstructed for three months, so the job must be completed in one month. How does this deadline
change the number of workers needed?
Hint 1. The proportionality
The time to complete the job should be inversely proportional to the number of workers on the
job. Therefore, reducing the time by a factor of 3 requires increasing the number of workers by a
factor of 3.
ANSWER:
One-ninth as many workers are needed.
One-third as many workers are needed.
The same number of workers are needed.
Three times as many workers are needed.
Nine times as many workers are needed.
Part F
The loudness of a sound is inversely proportional to the square of your distance from the source
of the sound. If your friend is right next to the speakers at a loud concert and you are four times
as far away from the speakers, how does the loudness of the music at your position compare to
the loudness at your friend's position?
ANSWER:
The sound is one-sixteenth as loud at your position.
The sound is one-fourth as loud at your position.
The sound is equally loud at your position.
The sound is four times as loud at your position.
The sound is sixteen times as loud at your position.
Inverse-square relations show up in the loudness of sounds, the brightness of lights, and the
strength of forces.
Proportional reasoning is useful for checking your answers to problems. If your answer is a
formula, then you can explicitly check that all of the variables have the correct proportionalities.
If you have a numerical answer, you can check your technique by doubling one of the starting
variables and working through the same process to a solution. If your answer does not change as
you expect it to based on the proportionality of the initial and final variables, then you know that
something is wrong.
Optimizing the Bakery
Learning Goal:
To use substitution and cancellation to solve a system of two linear equations in two unknowns.
Frequently, in physics, you will have two different unknown quantities (for instance, distance
and time). If you have two different equations involving these two unknown quantities, then you
will be able to find exact values for each of them. These will usually consist of two linear
equations—equations that only contain terms like 2x, y, and constant numbers—with two
unknowns. In this problem, you will learn how to solve such systems, as well as see how
equations of this type can be used to make a business more profitable.
Suppose that a baker makes cakes and cookies. He knows that he is most efficient when he
makes pairs of cakes (instead of one cake at a time) or a dozen dozens of cookies. Call the
number of pairs of cakes that he bakes in a week x and the number of grosses (dozens of dozens)
of cookies that he bakes in a week y.
There are two factors that limit how much he can bake in a week: He only wants to work for 40
hours a week and he only has one oven. Suppose that it takes the baker 1 hour to prepare a pair
of cakes or a gross of cookies (before they are placed in the oven). Since he only wants to work
40 hours a week, his output of pairs of cakes x and his output of grosses of cookies y are
constrained by the equation x + y =40.
To maximize the profit of the bakery, the first step is to find where the equations for all of the
constraints intersect. For the following part, you will look at x + y=40 and y=0, which is also a
constraint (specifically a minimum) since the baker cannot make a negative number of cookies.
Parts A and B might seem easier than most problems with linear systems, but in them you will
use the basic techniques needed to solve any linear system: adding equations to cancel variables
and substituting the value of one variable to find the value of the other.
Part A
One way of solving systems of linear equations is by adding a multiple of one equation to the
other. The multiplier for the first equation is chosen so that one of the two variables will cancel
out in the sum. What should you multiply the equation y=0 by so that when added to x + y = 40
the variable y will cancel out?
Express your answer numerically.
Hint 1. Consider another example
Suppose that you are trying to solve the system 5x + 2y = 10 and –x + y =3, and you wish to
cancel the variable y. In the first equation you have the term 2y. You can cancel this term by
adding the inverse, namely, -2y. How can you get a -2y term in the second equation? Since the y
term in the second equation is simply y, you must multiply the second equation through by -2.
This changes the second equation to 2x – 2y = -6. When you add this to original equation, the
variable y will not appear in the sum.
In this part, you are trying to cancel the term y from the first part. What is the inverse of y?
ANSWER:
-1
This leads to the sum
Thus, the point of intersection for the lines y =0 and x + y = 40 is (40,0). This point tells you that
if the baker makes no cookies, he can make (at most) 40 pairs of cakes in a week.
Part B
Which of the following shows the result of substituting x = 0 into 0.75x + 1.5y = 40?
Hint 1. How to approach the problem
Once you have a numeric value for one of the variables, it is usually easier to substitute that
value into one of the original equations and then solve for the other variable. In this case, you
know the value of x from the second equation. Replacing the x in the first equation with a zero
gives 0.75x + 1.5y = 40. After simplifying this equation, you can solve it to find the value of y.
Dividing both sides of this equation by 1.5 gives y = 40/1.5 = 80/3 ≈. This point tells you that, if
the baker makes no cakes, he can bake (at most) 26.7 grosses of cookies.
Part C
Next, you need to find the x coordinate of the point of intersection for the equations x + y =40
and 0.75x + 1.5y = 40. Multiply x = y = 40 by a number that will make the y’s cancel when you
add the two equations. Which of the following gives the resulting equation?
ANSWER:
When multiplying an equation so that you can cancel one of the variables, be certain to multiply
both sides of the equation. For instance, multiplying the above equation incorrectly would yield:-1.5x -1.5y = 40, which would give you very incorrect values when you tried to solve for x and y.
Part D
Now, add the two equations and find the value of x.
Express your answer to three significant figures.
ANSWER:
x= 26.7
Part E
Finally, substitute the value 26.7 (or the exact value of 80/3) for x in the first of the original
equations (x + y = 40) to find the value of y.
Once you have found the value of y using the first of the original equations, you can substitute
this value of y and the value of x(26.7) into the second of the original equations (0.75x + 1.5y =
40) to check your work. If the values that you find on the two sides of the equal sign are not
equal, then you need to recheck your work.
Express your answer to three significant figures.
ANSWER:
= 13.3
Suppose that this baker makes a profit of $20 on each pair of cakes and a profit of $25 on each
gross of cookies. Then his profit P is given by the formula P=20x +25y. To maximize a linear
function (such as this profit function) given linear constraints (such as those given in the
beginning of the problem) you only need to look at the points of intersection of the constraints.
In this case, the four points to check are (0,0), (40,0), (0,26.7), and (26.7,13.3). Plug the x and y
values from each point into the profit function to find which gives the greatest profit.
Part F
Find the solution of the system of equations 3x + 4y = 10 and x – y =1. Give the x value followed
by the y value, separated by a comma.
Express your answer in the form x,y.
Hint 1. Find the x value of the solution
Multiply the second equation by a number that will make the y terms cancel when the equation is
added to the first equation. Then solve for x in the resulting equation.
Express your answer as an integer.
Hint 1. Find the proper multiplier for x – y = 1.
You wish to cancel the 4y term from the first equation when you add the two equations (after
multiplying the second by some constant). The inverse of 4y is -4y. What do you need to
multiply –y by to get a -4y term in the second equation?
Express your answer as an integer.
ANSWER:
4
Now, find the sum of the equations 3x + 4y = 10 and 4x -4y = 4 and solve for x in the resulting
equation.
ANSWER:
x= 2
Now, substitute 2 for x in one of the two original equations and solve for y in the resulting
equation.
ANSWER:
x,y= 2, 1
Pay Up!
Learning Goal:
To learn to solve linear equations.
Almost every topic in physics will require you to solve linear equations—equations that don't
contain any higher powers of the variable such as x2, x3, etc. Linear equations are the simplest
algebraic equations. They arise in all sorts of situations. For this problem we'll look at one that
might come up in your daily life.
Suppose that you and three friends go out to eat and afterward decide to split the cost evenly.
Your friend Anika points out that she only had a drink, so she should pay less ($2, the cost of her
drink) and the rest of you can split the remainder of the bill. A linear equation can easily
determine how much each of you must pay.
For the particular problem raised in the introduction, assume that the total bill is $44. To answer
the question "How should the bill be split?" we will create a linear equation. The unknown is
how much money a single person (besides Anika) must pay, so call that x. Although four people
(you plus three friends) went to dinner, only three are paying the unknown amount x for a total
of 3x. Since Anika is paying $2, the total amount paid is 3x + 2 dollars, which must equal the
amount of the bill, $44. Thus, the equation to find is 3x + 2 = 44.
The steps for solving a linear equation are as follows:
1. Move all of the constants to the right side.
2. Move all of the variable terms (terms containing ) to the left side.
3. Divide both sides by the coefficient of the variable to isolate the variable.
You will go through these steps one at a time to solve the equation and determine how much
each person should pay.
Part A
The first step in solving a linear equation is moving all of the constants (i.e., numbers like 2 and
44 that aren't attached to an x) to the right side. What is the final value on the right side once
you've moved all of the constants?
Express your answer as an integer.
Hint 1. How to "move" a constant
The term move is not exactly an accurate description of how you get rid of the constant on one
side of an equation. What you actually do is add the opposite of that number to both sides. In this
way, the constant no longer appears on the side that you don't want it on.
For instance, if you have x + 3 =4, you want to move the 3 to the right. The opposite of 3 is -3,
so you would subtract 3 from both sides:
,
which reduces to
.
Similarly, if you had 2x – 12 = 3, you would add 12 to each side, because the opposite of -12 is
12, and so
,
.
ANSWER:
3x= 42
Part B
Now that you have 3x = 42, you need to isolate the variable so that you have an equation of the
form "x= some number." What is the value of x (i.e., the amount you must pay)?
Express your answer as an integer number of dollars.
Hint 1. How to isolate the variable
To remove the coefficient from the variable, simply divide both sides of the equation by that
coefficient. For instance, if you had 2x = 6, then you would divide both sides by 2:
,
yielding
.
ANSWER:
x= 14
The next problem looks more intimidating, but it requires the same procedures: Move all
constants to the right, then move all variables to the left, and finally divide both sides by the
variable’s coefficient.
Part C
If 13x – 23 = 4x + 22, what is the value of x?
Express your answer as an integer.
Hint 1. Collect the constant terms
You have the equation 13x – 23= 4x + 22 and need to get all of the constants on the right side.
Which of the following would get all of the constant terms on the right side of the equation?
ANSWER:
Add 23 to both sides.
Subtract 23 from both sides.
Add 22 to both sides.
Subtract 22 from both sides.
Hint 2. Collect the terms with x
After adding 23 to both sides, you have the equation 13x = 4x + 45 and need to get all of the
terms with x on the left side. Which of the following would get all of the variable terms on the
left side?
ANSWER:
Add 13x to both sides.
Subtract 13x from both sides.
Add 4x to both sides.
Subtract 4x from both sides.
After subtracting 4x from both sides of the equation, you are left with 9x = 45. Dividing both
sides by the coefficient (9) of x will give you the value of x. Remember that you must always
add and subtract to get all of the constant terms on the right side and all of the variable terms on
the left side before multiplying or dividing to isolate the variable.
ANSWER:
x= 5
Part D
If 5x – 9 = -2x -19, what is the value of x?
Express your answer as an integer.
Hint 1. Collect the constant terms
You have the equation 5x – 9 = -2x -19 and need to get all of the constants on the right side.
Which of the following would get all of the constant terms on the right side of the equation?
ANSWER:
Add 9 to both sides.
Subtract 9 from both sides.
Add 19 to both sides.
Subtract 19 from both sides.
Hint 2. Collect the terms with x
After subtracting 9 from both sides, you have the equation 5x= -2x -28.You need to get all of the
terms with x on the left side. Which of the following would get all of the variable terms on the
left side?
ANSWER:
Add 5x to both sides.
Subtract 5x from both sides.
Add 2x to both sides.
Subtract 2x from both sides.
After adding 2x to both sides of the equation, you are left with 7x = -28. Dividing both sides by
the coefficient (7) of x will give you the value of x. Remember that you must always add and
subtract to get all of the constant terms on the right side and all of the variable terms on the left
side before multiplying or dividing to isolate the variable.
ANSWER:
x= -4
Solving Linear Equations
Learning Goal:
To learn how to solve and check linear equations in one unknown.
Algebra may seem puzzling, but this problem will show that solving linear equations comes
down to arithmetic.
Before trying to solve an equation, you would like to have an idea of what the solution should be,
so that the solution can be checked after solving. An equation can always be turned into a
question in words, which highlights the arithmetic nature of the problem. Rephrasing an equation
in words also makes it easier to estimate the value of the solution. For instance, solving the
equation 4x + 8 =60 is the same as answering the question "What number can you multiply by 4
and then add 8 to the product to get 60?"
Part A
Consider the numbers 0, 10, 20, 30, and 40. Multiply each by 4 and compare the result to 60 to
determine into which of the following intervals the answer to the question "What number can
you multiply by 4 and then add 8 to the product to get 60?" should fall.
Select the interval into which the answer should fall.
ANSWER:
0 to 10
10 to 20
20 to 30
30 to 40
Part B
The answer here, x = 7, is not in the interval that you selected in the previous part. What is
wrong with the work shown above?
ANSWER:
In the first step, both terms on the left should be divided by 4.
60/4 should yield 16, not 15.
The last step should be to add 8 rather than subtract 8.
The last step should be to subtract 15 from each side.
Remember that when solving equations you are performing actions on both sides of the equality,
not on individual elements on each side. Think "Subtract 2 from both sides" rather than "I'll
subtract 2 from here, and from over here." The distinction isn't obvious when you add or subtract,
but it is with multiplying and dividing. Notice "I'll divide both sides by 4" implies the correct
action: dividing each term on both sides by 4. If you just think "I need to divide here and here,"
you are more likely to make errors such as the one just demonstrated.
Part C
Recall that in Part A the equation 4x + 8 =60 was restated in words as, "What number multiplied
by 4 and then added to 8 gives 60?" Which of the following would be an equivalent way to write
this?
ANSWER:
What number can you multiply by 4 to get a number 8 less than 60?
What number can you subtract 60 from and then divide by 4 to get 8?
What number can you multiply by 4 to get a number 8 larger than 60?
What number can you divide by 4 to get a number 8 less than 60?
Part D
You know that a number 8 less than 60 is the number 52. So, you could rewrite your answer
from the previous part as, "What number can you multiply by 4 to get 52?" This is just the
definition of 52 divided by 4! So, what is the number x you are looking for?
Express your answer as an integer.
ANSWER:
x = 13
You've successfully solved this equation using simple arithmetic, and your answer falls into the
interval that you selected in Part A. What you just did was the same as following more formal
algebraic operations. Namely, Part C corresponds to the step
and Part D corresponds to the step
When you run into algebraic equations to solve, instead of trying to remember a list of rules, just
think about the arithmetic. Then, you will naturally arrive at the correct steps to manipulate the
equation to find a solution for x.
Solving Quadratic Equations
Learning Goal:
To use the quadratic formula to solve quadratic equations.
Maria wants to plant a small tomato garden in her yard. She bought 25 tomato plants, and she has
read that ideally tomatoes are planted in a square grid
to help them pollinate each other.
Part A
How many plants x should she plant in each row so that her 25 plants end up in a square (i.e., x
plants in each of x rows)?
Express your answer as an integer.
Hint 1. How to approach the problem
If the gardener has x plants in each of her x rows, then the total number of plants will be x2.
Since there are 25 total plants, the correct value of x will satisfy the equation x2 = 25. Solve this
equation by taking the square root of both sides. Note that 25 has both a positive and a negative
square root. Since you can't have a negative number of plants, you only want the positive square
root.
ANSWER:
5
You may have been able to solve this part simply by intuition or with simple arithmetic. In the
next part, you will use the quadratic formula to find the value of x. Although this involves more
work than necessary to solve this part, using the quadratic formula on a problem that you've
already solved should help you to feel comfortable with it.
Part B
To find x in Part A, you would need to solve the equation x2 – 25 =0. Which of the following
shows the proper values in the quadratic formula before simplifying the radical and dividing?
Hint 1. The values of a, b, and c.
A is the value of the coefficient of x2, so in this problem, a = 1 (because x2 = 1*x2). bis the value
of the coefficient of x. Notice that the equation given has no x term. Since 0*x=0, the coefficient
of x must be zero. Therefore, b=0. Finally, c is the constant term, so in this problem, c = -25.
Plug these values into the quadratic formula and then reduce it to the form shown in one of the
answer choices.
ANSWER:
Notice that the value of c is -25, not +25. You should always include the sign in the values of a,
b, and c. Also, notice that the answer implies there are two solutions to Part A: x = 5 and x = -5.
Of course, you can't have -5 tomato plants, so for solving the practical problem of planting
tomatoes, the only correct solution is 5 plants per row.
Part C
Consider the equation 2x2 -3x -5 = 0. Plug the values for a, b, and c into the quadratic formula,
but do not simplify at all. Which of the following shows the proper substitution?
ANSWER:
Part D
Use the result from Part C to find the two solutions to the equation 2x2 -3x -5 =0.
Enter the two solutions separated by a comma. (The order is not important.)
ANSWER:
-1, 2.50
Also accepted: 2.50, -1
The Sum of Parts
When you add or subtract fractions, you must first find the least common multiple of the
denominators. (The denominator is the bottom part; the numerator is the top part.) To see why
this is, notice that
.
Think of a fifth as one of five equal-sized pieces that a stick has been broken into. If you add 1
piece and 3 pieces, you have 1 + 3 pieces. If the denominators are different, then this cannot be
done. Therefore, you will have to get the two fractions to have the same denominator: the least
common multiple of the two individual denominators.
Part A
What is the least common multiple of the denominators for the two fractions 1/6 and 5/8?
Express your answer as an integer.
Hint 1. What does least common multiple mean?
To understand what least common multiple means, break down the term. A common multiple of
two numbers is a third number that can be divided by either of the first two (i.e., it is a multiple
of each). For instance, 12 is a common multiple of 2 and 3, since 12 = 2 * 6 and 12 = 3 * 4. The
least common multiple is the smallest number that can be divided by the two original numbers.
In the example here, 12 is not the least common multiple, because 6 is smaller and can be
divided by either 2 or 3.
If you use a common multiple that is not the least common multiple as your denominator, then
you will have to reduce the fraction at the end of your calculations. You should always check to
see whether you can reduce the fraction at the end, so don't worry too much about finding the
least common multiple if it gives you trouble, but be sure to find a common multiple.
ANSWER:
24
Part B
Find the sum of 1/6 and 5/8.
Enter the numerator followed by the denominator separated by a comma.
Hint 1. Convert 1/6
Find the numerator of 1/6 after it has been converted to a fraction with a denominator of 24.
Express your answer as an integer.
ANSWER:
1/6= 4 /24
Hint 2. Convert 5/8
Find the numerator of 5/8 after it has been converted to a fraction with a denominator of 24. You
can divide 24 by 8 to find the needed multiplicative factor.
Express your answer as an integer.
ANSWER:
5/8= 15 /24
ANSWER:
1/6 + 5/8= 19/24
Part C
Find the value of 5/12 – 1/7.
Enter the numerator followed by the denominator separated by a comma.
Hint 1. Find the least common multiple
What is the least common multiple of 7 and 12?
Express your answer as an integer.
ANSWER:
84
Hint 2. Convert 5/12
Find the numerator of 5/12 after it has been converted to a fraction with a denominator of 84.
Express your answer as an integer.
ANSWER:
5/12= 35 /84
Hint 3. Convert 1/7
Find the numerator of 1/7 after it has been converted to a fraction with a denominator of 84.
Express your answer as an integer.
ANSWER:
1/7= 12 /84
ANSWER:
5/12 – 1/7= 23/84
Part D
Find the sum of 8/9 and 3/4.
Enter the numerator followed by the denominator separated by a comma.
Hint 1. Find the least common multiple
What is the least common multiple of 9 and 4?
Express your answer as an integer.
ANSWER:
36
Hint 2. Convert 8/9
Find the numerator of 8/9 after it has been converted to a fraction with a denominator of 36.
Express your answer as an integer.
ANSWER:
8/9 = 32 /36
Hint 3. Convert 3/4
Find the numerator of 3/4 after it has been converted to a fraction with a denominator of 36.
Express your answer as an integer.
ANSWER:
3/4= 27 /36
ANSWER:
8/9 + ¾ = 59/ 36
Fractions with the numerator larger than the denominator are called "improper fractions," but as
far as physics is concerned there's nothing "improper" about them. In fact, they are usually more
useful than the mixed numbers (e.g., 2&1/2) you might convert them to.
Trig Functions and Right Triangles
Learning Goal:
To use trigonometric functions to find sides and angles of right triangles.
The functions sine, cosine, and tangent are called trigonometric functions (often shortened to
"trig functions"). Trigonometric just means "measuring triangles." These functions are called
trigonometric because they are used to find the lengths of sides or the measures of angles for
right triangles. They can be used, with some effort, to find measures of any triangle, but in this
problem we will focus on right triangles. Right triangles are by far the most commonly used
triangles in physics, and they are particularly easy to measure.
The sine, cosine, and tangent functions of an acute angle in a right triangle are defined using the
relative labels "opposite side" O and "adjacent side" A.
The hypotenuse H is the side opposite the right
angle.
As you can see from the figure, the opposite side O is the side of the triangle not involved in
making the angle. The side called the adjacent side A is the side involved in making the angle
that is not the hypotenuse. (The hypotenuse will always be one of the two sides making up the
angle, because you will always look at the acute angles, not the right angle.)
The sine function of an angle θ, written sin(θ), is defined as the ratio of the length O of opposite
side to the length H of the hypotenuse:
.
You can use your calculator to find the value of sine for any angle. You can then use the sine to
find the length of the hypotenuse from the length of the opposite side, or vice versa, by using the
fact that the previous formula may be rewritten in either of the following two forms:
or
.
Part A
Suppose that you need to get a heavy couch into the bed of a pickup truck.
You know the bed of the truck is at a height
of 1.00 m and you need a ramp that makes an angle of 40 with the ground if you are to be able
to push the couch.
Use the sine function to determine how long of a board you need to use to make a ramp that just
reaches the 1.00-m high truck bed at a 40 angle to the ground.
Express your answer in meters to three significant figures.
Hint 1. Using the sine function
The ramp is the hypotenuse of the right triangle in the figure, and the side of length 1.00 m is
opposite the 40 angle. To find the length of the hypotenuse, use the
form of the sine formula. Plugging in the given values will give you the length of the hypotenuse.
ANSWER:
1.56
Part B
You need to set up another simple ramp using the board from Part A (i.e., a board of length 1.56
m).
If the ramp must be at a 25 angle above
the ground, how far back from the bed of the truck should the board touch the ground? Assume
this is a different truck than the one from Part A.
Express your answer in meters to three significant figures.
Hint 1. Using the cosine function
The ramp is the hypotenuse of the right triangle in the figure, and the distance along the ground
is adjacent to the 25 angle. To find the length of the adjacent side, use the
form of the cosine formula. Plugging in the given values will give you the distance along the
ground.
ANSWER:
1.41
Part C
Surveyors frequently use trig functions. Suppose that you measure the angle from your position
to the top of a mountain to be 2.50 .
If the
mountain is 1.00 km higher in elevation than your position, how far away is the mountain?
Express your answer in kilometers to three significant figures.
Hint 1. Using the tangent function
The height of the mountain is opposite the 2.50 angle of the right triangle in the figure, and the
distance to the mountain is adjacent to the 2.50 angle. To find the distance to the mountain, use
the
form of the tangent formula. Plugging in the given values will give you the distance to the
mountain.
ANSWER:
22.9
All of the trig functions also have inverses. The inverses of the sine, cosine, and tangent
functions are written as sin-1, cos-1, and tan-1, respectively. [Be careful not to confuse the notation
sin-1(x) for the inverse sine function with (sin(x))-1 = 1/sin(x).] These inverse functions are also
sometimes written asin, acos, and atan, short for arcsine, arccosine, and arctangent, respectively.
Your calculator should have three buttons with one of those sets of three labels.
Since a trig function takes an angle and gives a ratio of sides, the inverse trig functions must take
as input a ratio of sides and then give back an angle. For example, if you know that the length of
the side adjacent to a particular angle θ is 12 cm and the length of the hypotenuse of this triangle
is 13 cm, you can find the measure of angle θ using the inverse cosine. The cosine of θ would be
12/13, so the inverse cosine of 12/13 will give the value of θ:
implies that
.
Using the cos-1 or acos button on your calculator, you should check that the measure of θ is 22.6
.
Part D
The 3-4-5 right triangle is a commonly used right triangle.
Use the inverse sine function to determine the
measure of the angle opposite the side of length 3.
Express your angle in degrees to three significant figures.
Hint 1. Using the inverse sine
To use the inverse sine, first write down the formula for the sine of the angle:
for the triangle in the figure. This tells you that the measure of the angle θ is the inverse sine of
3/5.
ANSWER:
θ= 36.9 degrees
Part E
A support wire is attached to a recently transplanted tree to be sure that it stays vertical. The wire
is attached to the tree at a point 1.50 m from the ground, and the wire is 2.00 m long.
What is the angle ϕ between the tree and the
support wire?
Express your answer in degrees to three significant figures.
Hint 1. Choose the correct function
Using the given information, which of the following functions should you use to find the
measure of ϕ?
ANSWER:
ANSWER:
ϕ= 41.4 degrees
Understanding Functions
Learning Goal:
To understand and be able to use the idea of a function.
A function is a rule for relating each object in one set (the domain) to one, and only one, object
in another set (the range). For instance, we can create a function, call it FavoriteColor, that
relates each student in a class to a color. For this function, the domain is the set of students in the
class and the range is the set of colors.
It will be helpful to break down the definition of a function given in the introduction:



"A rule" just means that we always do it the same way: If Susan's favorite color is red,
then FavoriteColor(Susan) will always equal red.
"Relating each object in one set" just means that the rule gives us something for
everything in the domain: Every student has a favorite color.
"To one, and only one, object in another set" means that when you apply a function to
some object you only get one object in return: Zach's favorite color can be any color, but
it cannot be two different colors.
The figure
shows the actions of the function
FavoriteColor (i.e., the arrows indicate which color each person is related to).
Part A
What is the value of FavoriteColor(Troy)?
ANSWER:
green
red
magenta
cyan
blue
yellow
Part B
If FavoriteColor(x)=magenta, what is the value of x?
Hint 1. What is this question asking?
The question asks for the value of x such that FavoriteColor(x)=magenta. You could rephrase
this as, "To what person would you apply the function FavoriteColor to get the result magenta?"
Both of these are just fancy ways of asking whose favorite color is magenta, but it is important to
understand these questions, because they ask you to think in terms of functions, which are very
important to understanding physics. You can determine whose favorite color is magenta from the
figure.
ANSWER:
Susan
Zach
Troy
Janelle
Beverly
You might think that talking about functions made asking the simple question "Whose favorite
color is magenta?" seem complex. Instead, think of this as a sign that things written in the
language of functions are not as complicated as they first appear. The language of functions
gives a very precise way of discussing things.
Also, notice that seemingly complicated equations are frequently asking relatively simple things.
A question like "Solve for x in x2 = 4" is really just the question, "What number multiplied by
itself gives 4?"
Part C
Using the graph, determine Janelle's favorite color.
Hint 1. How to use the graph
You wish to find the value of FavoriteColor(Janelle) (i.e., the color to which Janelle is related by
FavoriteColor). To use the graph, start on the horizontal axis at the point labeled "Janelle." Move
vertically until you encounter a dot, then move toward the vertical axis. The value found on the
vertical axis is the value of FavoriteColor(Janelle).
ANSWER:
red
green
magenta
cyan
blue
yellow
Part D
For how many values of x does FavoriteColor(x)=red?
Use the graph to answer.
Hint 1. How to use the graph
In this part, you will use the graph in the reverse direction that you did for Part C. Start on the
vertical axis at the point labeled "red." Moving horizontally, you will encounter the dots
corresponding to names (elements of the domain) that are related to red by FavoriteColor.
Moving vertically down to the names would tell you which values of x correspond to red
according to the FavoriteColor function. You are only interested in the total number of such
points. Therefore, you can just move across and count the number of dots that you encounter.
ANSWER:
2
Part E
According to the definition of a function, each object in the domain is related "to one, and only
one, object in another set (the range)." In the arrow diagram originally shown, this property
could be seen from the fact that each name in the domain had only one arrow attached to it. How
can this be seen from the graph?
ANSWER:
A horizontal line intersects no more than one point on the graph.
A vertical line intersects no more than one point on the graph.
The graph does not intersect itself.
The graph is a continuous, unbroken curve.
Part F
Many quantities in physics are functions of time. For instance, the position of an object is a
function of time. Using the criterion from the previous part, determine which of the following
cannot be a graph of position vs. time.
ANSWER:
A
B
C
D