HW 2.5 Applications Maximizing and Minimizing-Brandi Malone
Student: _____________________
Date: _____________________
1.
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Instructor: Brandi Malone
Course: MATH 1830
According to a study, a vehicle's fuel economy, in miles per
gallon (mpg), decreases rapidly for speeds over 60 mph.
Assignment: HW 2.5 Applications
Maximizing and Minimizing
36
a) Estimate the speed at which the absolute maximum
gasoline mileage is obtained.
b) Estimate the speed at which the absolute minimum
gasoline mileage is obtained.
c) What is the mileage obtained at 70 mph?
Fuel economy (mpg)
30
a) The speed at which the absolute maximum gasoline
mileage is obtained is approximately
mph.
24
18
12
(Type a whole number.)
6
b) The speed at which the absolute minimum gasoline
mileage is obtained is approximately
mph.
0
5
Speed (mph)
(Type a whole number.)
c) The mileage obtained at 70 mph is
15 25 35 45 55 65 75 85 95
mpg.
(Type a whole number.)
2. An employee's monthly productivity M, in number of units produced, is found to be a function of the number t of years of
service. For a certain product, a productivity function is shown below. Find the maximum productivity and the year in which it
is achieved.
2
M(t) = − 3t + 180t + 180, 0 ≤ t ≤ 40
The maximum productivity is achieved in year
The maximum productivity is
.
units.
3. The total-cost, C(x), and total-revenue, R(x), functions for producing x items are shown below, where 0 ≤ x ≤ 300.
C(x) = 6000 + 100x and R(x) = −
1 2
x + 200x
2
a) Find the total-profit function P(x).
b) Find the number of items, x, for which the total profit is a maximum.
a) P(x) =
b) The profit is maximized for a production of
units.
4. For a dosage of x cubic centimeters (cc) of a certain drug, the resulting blood pressure B is approximated by the function
below. Find the maximum blood pressure and the dosage at which it occurs.
2
3
B(x) = 360x − 3600x , 0 ≤ x ≤ 0.10
The maximum is obtained for a dosage of
.
(Round to two decimal places as needed.)
The maximum blood pressure obtained is
.
(Round to two decimal places as needed.)
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5. A firm has determined that its cost of service, C(x), in thousands of dollars, is modeled by the given function, where x
represents the number of "quality units." Find the number of "quality units" that the firm should use in order to minimize its
total cost of service.
C(x) = (4x + 5) +
4
x−6
, x>6
The firm has a minimum cost of $
at x =
"quality units."
(Simplify your answers.)
6.
The table to the right gives the pressure of a pregnant
woman's contractions as a function of time. Complete parts
a) and b) using a calculator that has the REGRESSION
option.
a) Fit a linear equation to the data. Predict the pressure of
the contractions after 9 min.
b) Fit a quartic polynomial to the data. Predict the pressure
of the contractions after 9 min. Find the smallest contraction
over the interval [0,10].
Time, t
Pressure
(in minutes) (in millimeters of mercury)
0
11
1
9
2
10
3
15
4
12.5
5
14
6
14.5
a) Find the linear equation of the regression. Choose the correct answer below.
A. y = 3.00x + 9.01
B. y = 9.82x + 0.82
C. y = 55.29x − 153.58
D. y = 0.82x + 9.82
The pressure of the contractions after 9 min is approximately
millimeters of mercury.
(Type an integer or decimal rounded to two decimal places as needed.)
b) Find the equation of the quartic polynomial. Choose the correct answer below.
A. y = 0.1061x4 − 1.384x3 + 5.712x2 − 6.764x + 11.034
B. y = 0.1061x2 − 1.384x + 5.712
C. y = 0.1061x − 1.384
D. y = 0.1061x3 − 1.384x2 + 5.712x − 6.764
The pressure of the contractions after 9 min is approximately
millimeters of mercury.
(Type an integer or decimal rounded to two decimal places as needed.)
The smallest contraction over the interval [0,10] is at
minute(s) with
millimeters of mercury.
(Round to the nearest whole number as needed.)
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7. Of all numbers x and y whose sum is 50, the two that have the maximum product are x = 25 and y = 25. That is, if
x + y = 50, then x = 25 and y = 25 maximize Q = xy. Can there be a minimum product? Why or why not?
Choose the correct answer below.
A. No, there cannot be a minimum product. Since Q′′(x) < 0 for all x, any critical value must correspond to
a maximum product.
B. Yes, there can be a minimum product. Since Q′′(x) > 0 for all x, there must be a minimum product.
C. Yes, there can be a minimum product. Since Q′′(x) < 0 for all x, there must be a minimum product.
D. No, there cannot be a minimum product. Since Q′′(x) > 0 for all x, any critical value must correspond to
a maximum product.
8. Of all numbers whose difference is 10, find the two that have the minimum product.
What are the two numbers?
(Use a comma to separate answers as needed.)
9. Maximize Q = xy2 , where x and y are positive numbers such that x + y2 = 1.
x=
y=
(Type an exact answer, using radicals as needed.)
10. Minimize Q = 4x2 + 3y2 , where x + y = 7.
x=
y=
(Simplify your answer. Type an exact answer, using radicals as needed. Use integers or fractions for any numbers in the
expression.)
11. A rectangular plot of farmland will be bounded on one side by a river and on the other three sides by a single-strand
electric fence. With 500 m of wire at your disposal, what is the largest area you can enclose, and what are its dimensions?
The maximum area of the rectangular plot is
(1)
The length of the shorter side of the rectangular plot is
The length of the longer side of the rectangular plot is
(1)
2
m .
m.
3
(3)
(3)
m.
3
m.
m .
m .
3
m .
2
m .
m .
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(2)
(2)
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12. A carpenter is building a rectangular shed with a fixed perimeter of 44 ft. What are the dimensions of the largest shed that
can be built? What is its area?
The dimensions of the largest shed are
(1)
The area of the largest shed is
(1)
(2)
ft
ft
ft .
ft
2
ft .
(2)
(3)
(3)
ft.
3
by
3
3
ft .
2
ft .
2
ft.
13. Find the dimensions of the open rectangular box of maximum volume that can be made from a sheet of cardboard 33 in.
by 18 in. by cutting congruent squares from the corners and folding up the sides. Then find the volume.
The dimensions of box of maximum volume are
(1)
(Round to the nearest hundredth as needed. Use a comma to separate answers as needed.)
The maximum volume is
(2)
(Round to the nearest hundredth as needed.)
(1)
2
(2)
in. .
3
in. .
2
in.
in. .
3
in. .
in.
14. Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit.
Assume that revenue, R(x), and cost, C(x), of producing x units are in dollars.
R(x) = 60x − 0.1x2 , C(x) = 4x + 20
In order to yield the maximum profit of $
,
units must be produced and sold.
(Simplify your answers. Round to the nearest cent as needed.)
15. Find the maximum profit and the number of units that must be produced and sold in order to yield the maximum profit.
Assume that revenue, R(x), and cost, C(x), of producing x units are in dollars.
2
R(x) = 3x, C(x) = 0.02x + 0.8x + 1
What is the production level for the maximum profit?
units
What is the profit?
$
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16. Raggs, Ltd. a clothing firm, determines that in order to sell x suits, the price per suit must be p = 200 − 0.5x. It also
2
determines that the total cost of producing x suits is given by C(x) = 4000 + 0.75x .
a) Find the total revenue, R(x).
b) Find the total profit, P(x).
c) How many suits must the company produce and sell in order to maximize profit?
d) What is the maximum profit?
e) What price per suit must be charged in order to maximize profit?
a) R(x) =
b) P(x) =
c)
suits
d) The maximum profit is $
.
e) The price per unit must be $
.
17. A university is trying to determine what price to charge for tickets to football games. At a price of $26 per ticket, attendance
averages 40,000 people per game. Every decrease of $2 adds 10,000 people to the average number. Every person at the
game spends an average of $5.00 on concessions. What price per ticket should be charged in order to maximize revenue?
How many people will attend at that price?
What is the price per ticket?
$
What is the average attendance?
people
18. A rectangular box with a volume of 784 ft3 is to be constructed with a square base and top.
The cost per square foot for the bottom is 15¢, for the top is 10¢, and for the sides is
1.5¢. What dimensions will minimize the cost?
What are the dimensions of the box?
The length of one side of the base is
The height of the box is
(1)
(2)
(Round to one decimal place as needed.)
(1)
3
ft .
2
5 of 10
(2)
3
ft .
2
ft .
ft .
ft.
ft.
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19. A Norman window is a rectangle with a semicircle on top. Suppose that the perimeter of a particular
Norman window is to be 26 ft. What should its dimensions be in order to allow the maximum amount of light
to enter through the window?
x
y
To allow the maximum amount of light to enter through the window, the radius of the semicircle should be
(1)
and the height of the rectangle should be
(2)
(Do not round until the final answer. Then round to two decimal places as needed.)
(1)
ft
(2)
3
ft .
ft
3
ft.
ft
2
ft .
2
20. The amount of money that customers deposit in a bank in savings accounts is directly proportional to the interest rate that
the bank pays on that money. Suppose that a bank was able to turn around and loan out all the money deposited in its
savings accounts at an interest rate of 23%. What interest rate should it pay on its savings accounts in order to maximize
profit?
The bank should pay
% on its savings accounts.
21.
A power line is to be constructed from a power station at point A to an
island at point C, which is 1 mi directly out in the water from a point B on the
shore. Point B is 4 mi downshore from the power station at A. It costs
$2800 per mile to lay the power line under water and $2000 per mile to lay
the line under ground. At what point S downshore from A should the line
come to the shore in order to minimize cost? Note that S could very well be
B or A. (Hint: The length of CS is
2
1 + x .)
1
x
4 −x
4
S is
miles from A.
(Round to two decimal places as needed.)
22.
How does increasing the Total Yearly Units affect the
total yearly cost and the optimal order size?
(Use the interactive figure to find your answer.)
Click here to launch the interactive figure.1
Choose the correct answer below.
A. Increasing the total yearly units increases total yearly cost and decreases the optimal order size.
B. Increasing the total yearly units increases total yearly cost but does not affect the optimal order size.
C. Increasing the total yearly units increases total yearly cost and increases the optimal order size.
1: http://media.pearsoncmg.com/aw/aw_bittinger_cwa10e_12/mif/cwa10b_ch2/bescalc10_2_5_1.nbp
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23.
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How does increasing the Price Per Item affect the order
size?
(Use the interactive figure to find your answer.)
Click here to launch the interactive figure.2
Choose the correct answer below.
A. Increasing the price per item decreases the order size.
B. Increasing the price per item does not affect the order size.
C. Increasing the price per item increases the order size.
2: http://media.pearsoncmg.com/aw/aw_bittinger_cwa10e_12/mif/cwa10b_ch2/bescalc10_2_5_1.nbp
24.
How does increasing the Storage Cost Per Item affect
the order size and the total yearly cost?
(Use the interactive figure to find your answer.)
Click here to launch the interactive figure.3
Choose the correct answer below.
A. Increasing the storage cost per item increases the order size and increases the total yearly cost.
B. Increasing the storage cost per item increases the order size and decreases the total yearly cost.
C. Increasing the storage cost per item decreases the order size and decreases the total yearly cost.
D. Increasing the storage cost per item decreases the order size and increases the total yearly cost.
3: http://media.pearsoncmg.com/aw/aw_bittinger_cwa10e_12/mif/cwa10b_ch2/bescalc10_2_5_1.nbp
25.
Is the following statement true or false?
An open-top box with a fixed volume will have equal surface area
regardless of width and height dimension.
(Use the interactive figure to find your answer.)
Click here to launch the interactive figure.4
Is the statement true or false?
True
False
4: http://media.pearsoncmg.com/aw/aw_bittinger_cwa10e_12/mif/cwa10b_ch2/bescalc10_2_5_2.nbp
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1. 55
5
24
2. 30
2880
3.
−
1 2
x + 100x − 6000
2
100
4. 0.07
0.53
5. 37,000
7
6. D. y = 0.82x + 9.82
17.21
4
3
2
A. y = 0.1061x − 1.384x + 5.712x − 6.764x + 11.034
99.88
1
9
7. A.
No, there cannot be a minimum product. Since Q′′(x) < 0 for all x, any critical value must correspond to a maximum
product.
8. 5, − 5
9. 1
2
2
2
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10. 3
4
11. 31,250
2
(1) m .
125
(2) m.
250
(3) m.
12. 11
(1) ft
11
(2) ft.
121
2
(3) ft .
13. 3.73,10.54,25.54
(1) in.
1004.08
3
(2) in. .
14. 7820.00
280
15. 55
59.50
16. 200x − 0.5x2
2
− 1.25x + 200x − 4000
80
4,000.00
160.00
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17. 14.50
97,500
18. 4.5
(1) ft.
37.9
(2) ft.
19. 3.64
(1) ft
3.64
(2) ft.
20. 11.5
21. 2.98
22. C. Increasing the total yearly units increases total yearly cost and increases the optimal order size.
23. B. Increasing the price per item does not affect the order size.
24. D. Increasing the storage cost per item decreases the order size and increases the total yearly cost.
25. False
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