Applications of Functions
1.
Two numbers add to 5. What is the largest possible value of their product?
2.
Find two numbers adding to 20 such that the sum of their squares is as small as
possible.
3.
The difference of two numbers is 1. What is the smallest possible value for the sum
of their squares?
4.
For each quadratic function specified below, state whether it would make sense to
look for a highest or a lowest point on the graph. Then determine the coordinates of
that point.
a. y = 2x2 – 8x + 1
b. y = –3x2 – 4x – 9
c. h = –16t2 + 256t
d. f(x) = 1 – (x+1)2
e. g(t) = t2 + 1
f. f(x) = 1000x2 – x + 100
5.
Among all rectangles having a perimeter of 25 m, find the dimensions of the one
with the largest area.
6.
What is the largest possible area for a rectangle whose perimeter is 80 cm?
7.
What is the largest possible area for a for a right triangle in which the sum of the
lengths of the two shorter sides is 100 in?
8.
The perimeter of a rectangle is 12 m. Find the dimensions for which the diagonal is
as short as possible.
9.
a)
b)
c)
d)
e)
f)
g)
h)
i)
j)
Math 151
Minimize S = 6x2 – 2xy + 5y2 given that x + y = 13.
Minimize S = 12x2 + 4xy – 10y2 given that x + y = 14.
Maximize S = 3x2 + 5xy – 2y2 given that x + y = 8.
Maximize S = 4x2 + 3xy – 5y2 given that x + y = –8.
Maximize S = –2x2 + 3xy – 5y2 given that x + y = 20.
Minimize S = 3x2 + 2xy + 2y2 given that 3x – 2y = 42.
Minimize S = 2x2 + 5xy – y2 given that 4x – y = 12.
Minimize S = 3x2 – xy + 2y2 given that x – 2y = 24.
Maximize S = –3x2 + 2xy + 4y2 given that 2x – 5y = –9.
Maximize S = –x2 + 6xy – 7y2 given that 2x – 3y = 2.
1
Applications of Functions
10.
Two numbers add to 6.
a. Let T denote the sum of the squares of the two numbers. What is the smallest
possible value for T?
b. Let S denote the sum of the first number and the square of the second. What is
the smallest possible value for S?
c. Let U denote the sum of the first number and twice the square of the second
number. What is the smallest possible value for U?
d. Let V denote the sum of the first number and the square of twice the second.
What is the smallest possible value for V?
11.
Suppose that the height of an object shot straight up is given by h(t) = 512t  16t2 (h
in feet and t in seconds). Find the maximum height and the time at which the object
hits the ground.
12.
A baseball is thrown straight up, and its height as a function of time is given by the
formula h(t) = –16t2 + 32t (h in feet and t in seconds).
a. Find the height of the ball when t = 1 and when t = 32 .
b. Find the maximum height of the ball and the time at which that height is
attained.
c. At what time(s) is the height 7 feet?
13.
What point is nearest to (3, 0) on the curve y =
x ?
14.
What point is closest to (4, 1) on the curve y =
x – 2 + 1?
15.
Find the coordinates of the point on the line y = 3x + 1 closest to (4, 0).
16.
a. What number exceeds its square by the greatest amount?
b. What number exceeds twice its square by the greatest amount?
Suppose that you have 1800 meters of fencing available with which to build three
adjacent rectangular corrals as shown in the figure. Find the
dimensions so that the total enclosed area is as large as possible.
17.
18.
Five hundred feet of fencing is available for a rectangular pasture
alongside a river, the river serving as one side of the rectangle (so only
three sides require fencing). Find the dimensions yielding the greatest
area.
Problem 17
19.
Let A = 3x2 + 4x – 5 and B = x2 – 4x – 1. Find the minimum value of A – B.
20.
Let R = 0.4x2 + 10x + 5 and C = 0.5x2 + 2x + 101. For which value of x is R – C a
maximum?
21.
Suppose that the revenue generated by selling x units of a certain commodity is
given by R = – 15 x2 + 200x. Assume that R is in dollars. What is the maximum
revenue possible in this situation?
Math 151
2
Applications of Functions
22.
Suppose that the function p(x) = – 14 x + 30 relates the selling price p of an item to
the quantity sold, x. Assume P is in dollars. For which value of x will the
corresponding revenue be a maximum? What is this maximum revenue and what is
the unit price in this case?
23.
A piece of wire 200 cm long is to be cut into two pieces of lengths x and 200 – x.
The first piece is to be bent into a circle and the second piece into a square. For
which value of x is the combined area of the circle and square as small as possible?
24.
A 30 in piece of string is to be cut into two pieces. The first piece will be formed into
the shape of an equilateral triangle and the second piece into a square. Find the
length of the first piece if the combined area of the triangle and the square is to be
as small as possible?
25.
a. Same as exercise 23 except both pieces are to be formed into squares.
b. Could you have guessed the answer to part a?
26.
The action of sunlight on automobile exhaust produces air pollutants known as
photochemical oxidants. In a study of cross-country runners in Los Angeles, it was
shown that running performances can be adversely affected when the oxidant level
reaches 0.03 parts per million. Let us suppose that on a given day the oxidant level
L is approximated by the formula
L = 0.059t2 – 0.354t + 0.557 (0  t  7).
Here, t is measured in hours, with t = 0 corresponding to 12 noon, and L is in parts
per million. At what time is the oxidant level L a minimum? At this time, is the
oxidant level high enough to affect a runner's performance?
27.
If x + y = 1, find the largest possible value of the quantity x2 – 2y2.
28.
a.
29.
Through a type of chemical reaction known as autocatalysys, the human body
produces the enzyme trypsin from the enzyme trypsinogen. (Trypsin then breaks
down proteins into amino acids, which the body needs for growth.) Let r denote the
rate of this chemical reaction in which trypsin is formed from trypsinogen. It has
been shown experimentally that r = kx(a – x), where r is the rate of the reaction, k is
a positive constant, a is the initial amount of trypsinogen, and x is the amount of
trypsin produced (so x increases as the reaction proceeds). Show that the reaction
Find the smallest possible value of the quantity x2 + y2 under the restriction that
2x + 3y = 6.
b. Find the radius of the circle whose center is at the origin and that is tangent to
the line 2x + 3y = 6. How does this answer relate to your answer in part a?
rate r is a maximum when x = a2 . In other words, the speed of the reaction is the
greatest when the amount of trypsin formed is half of the original amount of
trypsinogen.
30.
a. Let x + y = 15. Find the minimum value of the quantity x2 + y2.
b. Let C be a constant and x + y = C. Show that the minimum value of x 2 + y2 is
c2
2
Math 151
. Then use this result to check your answer in part a.
3
Applications of Functions
31.
Suppose that A, B, and C are positive constants and that x + y = C. Show that the
BC
AC
minimum value of Ax2 + By2 occurs when x =
and y =
.
A+B
A+B
32.
The figure at the right shows two concentric squares. For which
value of x is the shaded area a maximum?
33.
Find the largest value of the function f(x) =
1
.
x4 – 2x2 + 1
Problem 32
34.
A rancher, who wishes to fence off a rectangular area, finds that the fencing in the
east-west direction will require extra reinforcement due to the strong prevailing
winds. Because of this, the cost of fencing in the east-west direction will be $12 per
linear yard, as opposed to a cost of $8 per yard for fencing in the north-south
direction. Find the dimensions of the largest possible rectangular area that can be
fenced for $4800.
35.
Let f(x) = (x – a)2 + (x – b)2 + (x – c)2, where a, b, and c are constants. Show that
f(x) will be a minimum when x is the average of a, b, and c.
36.
Let y = a1(x – x1)2 + a2(x – x2)2, where a1, a2, x1, and x2 are all constants. Further,
suppose that a1 and a2 are both positive. Show that the minimum of this function
a1 x1 + a2 x2
occurs when x =
a1 + a 2 .
37.
Among all rectangles with a given perimeter P, find the dimensions of the one with
the shortest diagonal.
38.
The figure at the right shows a rectangle
inscribed in a given triangle of base b and
height h. Find the ratio of the area of the
triangle to the area of the rectangle when the
area of the rectangle is a maximum.
h
b
Problem 38
39.
a.
Find the coordinates of the point on the line y = mx + b that is closest to the
origin.
b. Find the perpendicular distance from the origin to the line y = mx + b.
c. Use part b. to find the perpendicular distance from the origin to the line
Ax + By + C = 0.
40.
The point P lies in the first quadrant on the graph of the line y = 7 – 3x. From the
point P, perpendiculars are drawn to the x axis and the y axis, respectively. What is
the largest possible area for the rectangle thus formed?
Math 151
4
Applications of Functions
41.
Find the largest possible area for the shaded rectangle shown in
b2
the figure is – 4m . Then use this to check your answer in
exercise 39.
42.
43.
y = mx + b
Show that the maximum possible area for a rectangle inscribed
in a circle of radius R is 2R2.
Problem 41
A Norman window is in the shape of a rectangle surmounted by a
semicircle, as shown in the figure. Assume that the perimeter of the
window is P, a constant. Find the values of h and r when the area is a
maximum and find this area.
r
h
Problem 43
44.
A rectangle is inscribed in a semicircle of radius 3. Find the largest possible area for
this rectangle.
45.
An athletic field with a perimeter of
1
4
mi consists of a
x
r
rectangle with a semicircle at each end, as shown in
the figure below. Find the dimensions x and r that
yield the greatest possible area for the rectangular
region.
Problem 45
46.
Find the values of x that make y a minimum or a maximum, as the case may be.
Find the corresponding y value and indicate whether it is a minimum or a maximum.
a. y = 3x4 – 12x2 – 5
b. y = 4x – x2
47.
By analyzing sales figures, the economist for a stereo manufacturer knows that 150
units of a top of the line turntable can be sold each month when the price is set at
p = $200 per unit. The figures also show that for each $10 hike in price, 5 fewer
units are sold each month.
a. Let x denote the number of units sold per month and let p denote the price per
unit. Find a linear function relating p and x.
b. Express the revenue R as a function of x.
c. What is the maximum revenue? At what level should the price be set to achieve
this maximum revenue?
48.
Imagine that you own an orchard of orange trees. Suppose from past experience
you know that when 100 trees are planted, each tree will yield approximately 240
oranges. Furthermore, you've noticed that when additional trees are planted, the
yield per (each) tree in the orchard decreases. Specifically, you have noted that the
yield per tree decreases by about 20 oranges for each additional tree planted.
Approximately how many trees should be planted in the orchard to produce the
largest possible total yield of oranges?
Math 151
5
Applications of Functions
49.
An appliance firm is marketing a new refrigerator. It determines that in order to sell
x refrigerators, its price per refrigerator must be p = D(x) = 280 – 0.4x. It also
determines that its total cost of producing x refrigerators is given by
C(x) = 5000 – 0.6x2.
a) How many refrigerators must the company produce and sell in order to
maximize profit?
b) What is the maximum profit?
c) What price per refrigerator must be charged in order to make this maximum
profit?
50.
The owner of a 30 unit motel find that all units are occupied when the charge is $20
per day per unit. For every increase of x dollars in the daily rate, there are x units
vacant. Each occupied room costs $2 per day to service and maintain. What
should he charge per unit per day in order to maximize profit?
51.
A university is trying to determine what price to charge for football tickets. At a price
of $6 per ticket, it averages $70,000 people per game. For every increase of $1, it
loses 10,000 people from the average number. Every person at the game spends
an average of $1.50 on concessions. What price per ticket should be charged in
order to maximize revenue? How many people will attend at that price?
52.
When a theater owner charges $3 for admission, there is an average attendance of
100 people. For every 10 cent increase in admission, there is a loss of 1 customer
from the average. What admission should be charged in order to maximize
revenue?
53.
An apple farm yields an average of 30 bushels of apples per tree when 20 trees are
planted on an acre of ground. Each time 1 more tree is planted per acre, the yield
decreases 1 bushel per tree due to the extra congestion. How many trees should
be planted in order to get the highest yield?
54.
A triangle is removed from a semicircle of radius R as shown in the
figure. Find the area of the remaining portion of the circle if it is to be
a minimum.
55.
Problem 54
Let f(x) = x2 + px + q, and suppose that the minimum value of this function is 0.
p2
Show that q = 4 .
56.
Suppose that x and y are both positive numbers and that their sum is 4. Find the
1
smallest possible value for the quantity xy
.
Math 151
6
Applications of Functions
Answers
1.
3.
6.25
5.
7.
9.
6.25 × 6.25
1250 in2
a) x = 6, y = 7, S = 377
1
2
b)
x = 4, y = 10, S = 952
c)
x = 9, y = –1, S = 196
d)
x = –13, y = 5, S = 356
e)
x = 13, y = 7, S = –310
f)
x = 8, y = –9
g)
x = –3, y = –24, S = –198
h)
x = 2, y = –11, S = 276
i)
x = 3, y = 3, S = 27
j)
x = 10, y = 6, S = 8
23
4
10.
a. 18; b.
; c.
11.
a. 16 ft, 12 ft
b. 16 ft at 1 sec
c. 14 sec, 74 sec
47
8
; d.
95
16
16.
7, 2+ 6 
2
2

1
1
a. 2 ; b. 4
18.
20.
22.
24.
26.
28.
125 (2 sides) × 250 ft
40
60; $900, $15
approximately 21.7
3 PM; no
a. 36
13
30.
a.
32.
1
2
34.
38.
40.
100 × 150
2
49
square units
12
41.
44.
46.
2R2
9 square units
a. (± 2 , –17)
b. min at (0, 0) and (4, 0); max at (2, 2)
56 trees
$26
$6.50
–2
R2  2 


14.
48.
50.
52.
54.
Math 151
S = 210
225
2
7
Applications of Functions