Deck 10: Optimization

Question 1

Maxima for functions can be found at peaks or at endpoints.

Answer 1

True

Question 2

Find the minimum value of .Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places. ​

A) 6.83
B) 0.83
C) 1.86
D) 10.47

Answer 2

6.83

Question 3

Find the maximum value of .Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places. ​

A) 2.50
B) 0.72
C) 4.14
D) 4.42

Answer 3

0.72

Question 4

What is the minimum value of ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places. ​

A) 2.09
B) 10.03
C) 0.75
D) 13.45

Question 5

For which value of x does reach a maximum? Restrict your attention to the horizontal span of 0 to 12.Round your answer to two decimal places. ​

A) 1.55
B) 3.17
C) 0.06
D) 7.09

Question 6

Assume that the function is increasing on the horizontal span of 2 to 25.Then the maximum value of on this horizontal span is .

Question 7

For what value of x does reach a minimum? Restrict your attention to the horizontal span of 0 to 11.Round your answer to two decimal places. ​

A) 2.42
B) 4.27
C) 0.78
D) 7.33

Question 8

For which value of x does reach a maximum? Restrict your attention to the horizontal span of 0 to 5.Round your answer to two decimal places. ​

A) 1.41
B) 15.89
C) 0.36
D) 19.55

Question 9

What is the maximum value of ? Restrict your attention to the horizontal span of 0 to 4.Round your answer to two decimal places. ​

A) 41.14
B) 2.92
C) 186.00
D) 1.78

Question 10

Minima always occur at endpoints.

Question 11

What is the maximum value of ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places, if necessary. ​

A) 8
B) 12.17
C) 0.83
D) 15.34

Question 12

Graphs can be used to find maxima and minima for functions.

Question 13

What is the maximum value of ? Restrict your attention to the horizontal span of 0 to 20.Round your answer to two decimal places. ​

A) 1.30
B) 4.58
C) 0.07
D) 7.66

Question 14

For which value of x does reach a minimum? Restrict your attention to the horizontal span of 0 to 5.Round your answer to two decimal places. ​

A) 13
B) 11.44
C) 0.63
D) 0

Question 15

What is the minimum value of ? Restrict your attention to the horizontal span of 0 to 20.Round your answer to two decimal places. ​

A) -74.69
B) 5.47
C) -70.70
D) 4.44

Question 16

No function can have both a maximum and a minimum value.

Question 17

What value of x gives the maximum value of ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places. ​

A) 2.86
B) 1.22
C) 4.72
D) 4.74

Question 18

What value of x gives the minimum value of ? Restrict your attention to the horizontal span of 0 to 19.Round your answer to two decimal places. ​

A) -3.65
B) 3.67
C) -0.17
D) 2.53

Question 19

Find the maximum value of .Restrict your attention to the horizontal span of 0 to 9.Round your answer to two decimal places. ​

A) 1.13
B) 20.06
C) 2.72
D) 23.89

Question 20

What is the minimum value of ? Restrict your attention to the horizontal span of 0 to 1.Round your answer to two decimal places. ​

A) 0.13
B) 1.48
C) 2.06
D) 5.46

Question 21

What is the minimum value of ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places. ​

A) 30.15
B) 6.11
C) 33.72
D) 4.22

Question 22

You put a drink in the freezer to cool.When it is chilled, you take the drink out but, being forgetful, leave it sitting on the kitchen counter.The temperature D, in degrees, of the drink t minutes after it is put in the freezer is given by ​ .
​
During the period from 5 minutes after it is put in the freezer until 20 minutes after, when is the drink the warmest, and what is its warmest temperature? Be careful.You are asked for the warmest temperature, not the coldest.Round your answers to two decimal places.
​

A) It reaches its warmest temperature of 45.13 degrees after 5.00 minutes.
B) It reaches its warmest temperature of 5.00 degrees after 45.13 minutes.
C) It reaches its warmest temperature of 20.00 degrees after 54.93 minutes.
D) It reaches its warmest temperature of 54.93 degrees after 20.00 minutes.

Question 23

One model for expenditures at high schools gives the cost C, in dollars per pupil, as a function of the number n of students enrolled.The relationship is . ​
Find the enrollment and cost that correspond to the minimum cost per pupil.Round your answers to the nearest whole number.

A) An enrollment of 1909 gives a minimum cost of $340 per pupil.
B) An enrollment of 340 gives a minimum cost of $1909 per pupil.
C) An enrollment of 1911 gives a minimum cost of $344 per pupil.
D) An enrollment of 344 gives a minimum cost of $1911 per pupil.

Question 24

The number N, measured in millions of students, enrolled in high school t years after 1965 is given approximately by ​ .
According to this model, in what year (as a date) did high school enroll reach a maximum, and what was the maximum enrollment? Round your answers to two decimal places.
​

A) The maximum enrollment of 13.68 million occurred in 1977.
B) The maximum enrollment of 12.00 million occurred in 1981.00.
C) The maximum enrollment of 13.44 million occurred in 1984.
D) There is no maximum value.

Question 25

For a certain epidemic the number N of new cases expected in a day depends on the number t of days since the outbreak began.The relationship is . ​
What is the greatest number of new cases that occur on a day, and on what day does that happen? Round both answers to the nearest whole number.

A) A maximum of 41 new cases occurs on day 24.
B) A maximum of 21 new cases occurs on day 38.
C) A maximum of 24 new cases occurs on day 41.
D) A maximum of 38 new cases occurs on day 21.

Question 26

What value of x gives a minimum for ? Restrict your attention to the horizontal span of 9 to 15.Be careful.You are asked about a minimum, not a maximum.Round your answer to two decimal places. ​

A) 47.69
B) 15.00
C) 37.26
D) 11.82

Question 27

You put a drink in the freezer to cool.When it is chilled, you take the drink out but, being forgetful, leave it sitting on the kitchen counter.The temperature D, in degrees, of the drink t minutes after it is put in the freezer is given by ​ .
​
When is the drink the coldest, and what is its coldest temperature? Round your answers to two decimal places.
​

A) It reaches its coldest temperature of 9.76 degrees after 25.76 minutes.
B) It reaches its coldest temperature of 25.76 degrees after 9.76 minutes.
C) It reaches its coldest temperature of 12.37 degrees after 29.30 minutes.
D) It reaches its coldest temperature of 29.30 degrees after 12.37 minutes.

Question 28

The circulation M, in thousands, of a magazine depends on the time t, in years, since initial publication.The following relationship is valid over the first 7.00 years: . When did magazine circulation reach a maximum, and what was the maximum circulation? Round your answers to two decimal places.
​

A) The maximum circulation of 4.67 thousand occurred 54.81 years after initial publication.
B) The maximum circulation of 54.81 thousand occurred 4.67 years after initial publication.
C) The maximum circulation of 2.89 thousand occurred 57.70 years after initial publication.
D) The maximum circulation of 57.70 thousand occurred 2.89 years after initial publication.

Question 29

What is the maximum value of ? Restrict your attention to the horizontal span of 0 to 20.Round your answer to two decimal places. ​

A) 39.13
B) 12.46
C) 42.72
D) 11.30

Question 30

You put a drink in the freezer to cool.When it is chilled, you take the drink out but, being forgetful, leave it sitting on the kitchen counter.The temperature D, in degrees, of the drink t minutes after it is put in the freezer is given by ​ .
​
When during the first 20 minutes is the drink the warmest, and what is its warmest temperature? Be careful.You are asked for the warmest temperature, not the coldest.Round your answers to two decimal places.
​

A) It reaches its warmest temperature of 72.00 degrees when it is first put in the freezer.
B) It reaches its warmest temperature of 0.00 degrees when it is first put in the freezer.
C) It reaches its warmest temperature of 20.00 degrees after 54.93 minutes.
D) It reaches its warmest temperature of 54.93 degrees after 20.00 minutes.

Question 31

What value of x gives the minimum value of ? Restrict your attention to the horizontal span of 0 to 10.Round your answer to two decimal places. ​

A) 29.38
B) 5.39
C) 4.05
D) 33.06

Question 32

A can is made from 6 square inches of aluminum.We can calculate both the height , in inches, and the volume V, in cubic inches, from the radius r, in inches, of the can: ​ Find the radius and height of the can of maximum volume.Round your answers to two decimal places.
​

A) The radius is 1.13 inches, and the height is 0.56 inches.
B) The radius is 3.34 inches, and the height is 1.83 inches.
C) The radius is 1.83 inches, and the height is 3.34 inches.
D) The radius is 0.56 inches, and the height is 1.13 inches.

Question 33

One model for the growth rate G, in thousands per year, of a certain species depends on the population N, in thousands.The relationship is ​ What population corresponds to the maximum growth rate, and what is that growth rate? Round your answers to two decimal places.
​

A) The maximum growth rate of 1.75 thousand per year corresponds to a population of 2.50 thousand.
B) The maximum growth rate of 2.50 thousand per year corresponds to a population of 1.75 thousand.
C) The maximum growth rate of 4.40 thousand per year corresponds to a population of 4.24 thousand.
D) The maximum growth rate of 4.24 thousand per year corresponds to a population of 4.40 thousand.

Question 34

What value of x gives a maximum for ? Restrict your attention to the horizontal span of 0 to 80.Round your answer to two decimal places. ​

A) 65.48
B) 19.82
C) 68.68
D) 18.23

Question 35

What is the minimum value for ? Restrict your attention to the horizontal span of 40 to 46.Be careful.You are asked about a minimum, not a maximum.Round your answer to two decimal places. ​

A) 179.02
B) 46.00
C) 175.66
D) 42.63

Question 36

You are laying phone cable between two cities and must cross a lake.You will lay L miles on land.The cost C, in dollars, depends on how much cable is on land and how much is in water.The relationship is . ​
How much cable do you lay on land to achieve a minimum cost, and what is that minimum cost? Round your answers to two decimal places.
​

A) The minimum cost of $7.25 is achieved if you lay 2543.73 miles of cable on land.
B) The minimum cost of $2543.73 is achieved if you lay 7.25 miles of cable on land.
C) The minimum cost of $5.32 is achieved if you lay 2541.63 miles of cable on land.
D) The minimum cost of $2541.63 is achieved if you lay 5.32 miles of cable on land.