Note: For a problem on the homework: 1 mile = 5280 feet.
Lesson 30
Variation:
- how one quantity changes (varies) in relation to another quantity
- quantities can vary directly, indirectly, jointly or combined
Examples of variation equations:
1.
Direct variation
Circumference of a circle varies directly as the length of
the diameter of the circle.
Ex. 1
Volume of a sphere is directly proportional as the cube of
the radius of the sphere
In direct variation, as the independent variable increases (decreases), so does the dependent
variable. In the above examples: As the diameter increases (decreases), so does the
circumference. As the radius increases (decreases), so does the volume.
For direct variation, think of form
where k is the constant of proportionality or
variation constant. Direct variation represents a constant times a variable or quantity.
2.
Indirect (inverse) variation
Time for a 100 mile trip varies inversely as
the rate of speed.
Ex. 2
The intensity of illumination is indirectly proportional as
the square of the distance from the light source.
In indirect or inverse variation, as the independent variable increases (decreases), the dependent
variable decreases (increases). In the above examples: As the rate of speed increases, the time
decreases (or vice versa). As the square of the distance increases (decreases), the illumination
decreases (increases).
For inverse variation, think of the form
where k is the constant of proportionality
or variation constant. Inverse variation represents a constant divided by a variable or
quantity.
3.
Joint variation (direct variation with more than one variable)
Distance varies jointly as the rate of speed and the time.
Volume of a cone is directly proportional as the product
Ex. 3 & 4
of the square of its radius and its height.
Area of a trapezoid varies jointly as the sum of its
bases and its height.
(continued on the next page)
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Note: For a problem on the homework: 1 mile = 5280 feet.
In joint variation, as the product of independent variables increases (decreases), so does the
dependent variable. In the above examples: As the product of rate of speed and time increases
(or decreases), the distance increases (or decreases). As the product of the radius squared and
the height increases (or decreases), so does the volume of the cone. As the product of the sum
of the bases and the height of the trapezoid increases (or decreases), so does the area of the
trapezoid.
For joint variation, think of the form
where k is the variation constant or the
constant of proportionality. Joint variation represents a constant multiplied by a product
involving variables. Joint variation is direct variation with more than one variable.
4.
Combined variation (both direct and indirect or inverse variation)
Length of a rectangular solid varies directly as its volume
and indirectly as the product of its width and height.
The number of watts of a resistor varies directly as the
square of the number of volts and inversely as the resistance.
For combined variation, think of the form
where k is the variation constant or the
constant of proportionality. Combined variation represents a constant multiplied by a
variable or a product involving variables and also divided by a variable (or a product
involving variables. Combined variation is both direct and inverse variation together.
Steps for solving a variation problem:
1.
Write a general variation equation that relates all variables and k, the variation
constant.
2.
Substitute some initially given values for the variables and solve for k.
3.
Write the specific variation equation. (Substitute k back in step 1.)
4.
Use the new variation equation to solve the problem.
Direct Variation:
- as the independent variable increases, the dependent variable also increases (as x
increases, so does y)
- Basic direct variation is described by the equation
, where is the dependent
variable, is the independent variable, is a nonzero constant
is known as the constant of variation or the constant of proportionality
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Note: For a problem on the homework: 1 mile = 5280 feet.
Example 1: Express the statement as a formula that involves the given variables and a constant
of proportionality . Find u when v = 12.
is directly proportional to . If
, then
.
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10
0
0
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As seen by the graph above, direct variation with a
single variable to the 1st power is represented by a
line graph. It is a linear equation. However, not all
direct variation graphs will be lines.
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Note: For a problem on the homework: 1 mile = 5280 feet.
Indirect Variation:
- as the independent variable increases, the dependent variable decreases
- basic indirect (or inverse) variation is described by the equation
, where
dependent variable, is the independent variable
- is known as the constant of variation or the constant of proportionality
is the
Same Steps for solving a variation problem:
1.
Write a variation equation that relates all variables and k, the variation constant.
2.
Substitute some given values for the variables and solve for k.
3.
Write the variation equation. (Substitute k back in step 1.)
4.
Use the variation equation to solve the problem.
Example 2: Express the statement as a formula that involves the given variables and a constant
of proportionality . Find p when q = 27.
is inversely proportional to the cube root of . If
, then
.
A note: Inverse or indirect variation
could be represented by a graph that
is a ‘curve’.
4
Note: For a problem on the homework: 1 mile = 5280 feet.
Joint Variation:
- if a quantity varies directly as the product of two or more variables, it is known as joint
variation
- basic joint variation is described by the equation
, where is the dependent
variable, and are the independent variables, is a nonzero constant
Example 3: Express the statement as a formula that involves the given variables and a constant
of proportionality .
is directly proportional to the product of the square root of
, then
.
and the cube of . If
Ex 4: q varies jointly with the square of m and the square root of n. If q = 45 when m =
and
3 and
n = 25, find q when m = 4 and n = 36.
5
Note: For a problem on the homework: 1 mile = 5280 feet.
Combined Variation:
- if a quantity varies directly (or jointly) with one or more variables and inversely with one
or more variables, it is known as combined variation
- basic combined variation can be described by the equation
, where is the
dependent variable,
, and are independent variables, is a nonzero constant This is
one example. There are many other examples.
Example 5: Express the statement as a formula that involves the given variables and a constant
of proportionality .
varies directly as the product of
, and
, then
.
and
, and inversely as the square of . If
,
Find F when Q1=5, Q2= 7, and d = 2.
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Note: For a problem on the homework: 1 mile = 5280 feet.
Example 6: An object’s weight on the moon varies directly as its weight on Earth .
a. Express the statement above as a formula that involves the given variables and a constant
of variation .
b. Neil Armstrong weighed 360 pounds on Earth (with all of his equipment) and 60 pounds
on the moon. Use this information to determine the value of .
c. What is the moon weight of a person who weighs 186 pounds on Earth?
The graph of the variation equation is a linear equation graphed below.
70 M
60
(360, 60)
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10
0
0
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420
E
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Note: For a problem on the homework: 1 mile = 5280 feet.
Ex 7: The time to drive a certain distance (t) varies inversely to the rate of speed (r). Mary drives 47 miles per
hour for 4 hours. How long would it take her to make the same trip at 55 mph? Round to the nearest tenth.
(Note: In webassign, only round answers when directed. Otherwise, give exact answers.)
Ex 8: The intensity of illumination on a surface varies inversely as the square of the distance from the light
source. A surface is 12 meters from a light source and has an intensity of 2. How far must the surface be from
the light source to receive twice as intense illumination? Approximate to the nearest tenth of a meter.
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Note: For a problem on the homework: 1 mile = 5280 feet.
Example 9: Radiation machines used to treat tumors produce an intensity of radiation that
varies inversely as the square of the distance from the machine.
a. Express the statement above as a formula that involves the given variables and a constant
of variation .
b. At 3 meters the radiation intensity is 62.5 mill roentgens per hour. Use this information
to determine the value of .
c. What is the intensity at 2 meters?
Below is the graph of this variation equation.
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0
0
0.5
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Note: For a problem on the homework: 1 mile = 5280 feet.
Example 10: The centrifugal force of a body moving in a circle varies jointly with the radius
of the circular path and the body’s mass , and inversely with the square of the time it takes
to move about one full circle.
a. Express the statement above as a formula that involves the given variables and a constant
of variation .
b. A 6-gram body moving in a circle with radius 100 centimeters at a rate of 1 revolution
every 2 seconds has a centrifugal force of 6000 dynes. Use this information to determine
the value of .
c.
Find the centrifugal force of an 18-gram body moving in a circle with radius 100
centimeters at a rate of 1 revolution every 3 seconds?
Ex 11:
The time to drive a certain distance (t) varies inversely to the rate of speed (r).
Mary drives 47 miles per hour for 4 hours. How long would it take her to make the same
trip at 55 mph? Round to the nearest tenth.
10
Note: For a problem on the homework: 1 mile = 5280 feet.
Ex 12:
The intensity of illumination on a surface varies inversely as the square of the
distance from the light source. A surface is 12 meters from a light source and has an
intensity of 2. How far must the surface be from the light source to receive twice as
much intensity of illumination? Approximate to the nearest tenth of a meter.
Ex13: The power, in watts, dissipated as heat in a resistor varies directly with the square of the
voltage and inversely with the resistance. If 20 volts are placed across a 20-ohm resistor,
it will dissipate 20 watts. What voltage across a 10-ohm resistor will dissipate 40 watts?
Ex 14:
You have heard of IQ, a person’s intelligence quotient. The value of IQ varies
directly as a person’s mental age and inversely as that person’s chronological age. A
person with a mental age of 30 and a chronological age of 25 has an IQ of 120. (a) What
is the chronological age of a person with a mental age of 30 and an IQ of 90? Round to
the nearest year, if necessary. (b) How about if a person’s mental age is 10 and the IQ is
120; estimate chronological age.
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Note: For a problem on the homework: 1 mile = 5280 feet.
Ex 15:
Volume V that a gas occupies varies directly as the product of the number of moles of gas (n) and the
temperature in K (T) and varies inversely as the pressure in atmospheres (P).
a)
Express V in terms of n, T, P, and a constant k.
b)
What is the effect on the volume if the number of moles is doubled, the temperature stays the same, and
the pressure is reduced by a factor of ½?
Ex 16:
Volume that a gas occupies varies directly as the product of the number of moles of gas (n) and the temperature
in K (T), and varies inversely to the pressure in atmospheres (P).
a)
Express V in terms of n, T, P, and a constant k.
b)
What is the effect on the volume if the number of moles is doubled, the temperature stays the same, and
the pressure is reduced by a factor of ½?
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