Deck 6: Analyzing Accumulated Change: Integrals in Action

Question 1

The average quantity of sculptures that consumers will demand can be modeled as and the average quantity that producers will supply can be modeled as where the market price is p hundred dollars per sculpture. By how much will demand exceed supply at a price of $500 per sculpture?

A) 581 sculptures
B) 183 sculptures
C) 600 sculptures
D) 417 sculptures
E) 784 sculptures

Answer 1

417 sculptures

Question 2

The daily demand for beef can be modeled by where the price for beef is p dollars per pound. Like-wise, the supply for beef can be modeled by where the price for beef is p dollars per pound. How much beef is supplied when the price is $3.50 per pound? Round your answer to the nearest million pounds.

A) 113 million pounds
B) 74 million pounds
C) 121 million pounds
D) 105 million pounds
E) 0 million pounds

Answer 2

113 million pounds

Question 3

Evaluate the improper integral if it converges, or state that it diverges.

A)
B)
C)
D)
E) diverges

Answer 3

diverges

Question 4

In 1956, AT&T laid its first underwater phone line. By 1996, AT&T Submarine Systems, the division of AT&T that installs and maintains undersea communication lines, had seven cable ships and 1000 workers. On October 5, 1996, AT&T announced that it was seeking a buyer for its Submarine Systems division. The Submarine Systems division of AT&T was posting a profit of $850 million per year. If a prospective bidder considered that over a 25-year period, profits of the division would grow by 10% per year (after which it would be obsolete) and that profits could be reinvested at an annual return of 20%, what would the prospective bidder have considered to be the 25-year present value of its Submarine Systems division? Assume a continuous stream and round your answer to the nearest billion dollars. You may use technology to calculate the answer.

A) $4 billion
B) $3 billion
C) $8 billion
D) $9 billion
E) $126 billion

Question 5

The willingness of saddle producers to supply saddles can be modeled by the following function: where saddles are sold for p thousand dollars. Find the producers' surplus when the market price is $7500. Round your answer to the nearest million dollars.

A) $9 million
B) $48 million
C) $39 million
D) $141 million
E) $6 million

Question 6

Between 1995 and 2001, the revenue of Sears Roebuck and Co. can be modeled as billion dollars per year, where t is the number of years after 1995. Assume that the revenue can be reinvested at 5.0% compounded annually. How much is Sears' revenue invested since 1995 worth in 2003? Round your answer to the nearest $10 billion.

A) $270 billion
B) $330 billion
C) $880 billion
D) $130 billion
E) $400 billion

Question 7

The demand for train sets can be modeled as train sets where p is the price (in dollars) of a train set. For what prices is demand elastic?

A) per train set
B) per train set
C) per train set
D) per train set
E) per train set

Question 8

The demand for circus tickets can be modeled as hundred tickets where p is the price (in dollars) of a ticket. According to the model, at what price will consumers no longer purchase circus tickets?

A) $23.50 per ticket
B) $94.00 per ticket
C) $4.04 per ticket
D) $117.50 per ticket
E) $8.00 per ticket

Question 9

The demand for train sets can be modeled as train sets where p is the price (in dollars) of a train set. Find the point of unit elasticity.

A) $5.58 per train set
B) $1.60 per train set
C) $29.38 per train set
D) $67.66 per train set
E) $2.42 per train set

Question 10

There are approximately 200 thousand northern fur seals. Suppose the population is being renewed at a rate of thousand seals per year and that the survival rate is 78% How many of the current population of 200 thousand seals will still be alive 25 years from now?

A) 244 seals
B) 401 seals
C) 514 seals
D) 201 seals
E) 156 seals

Question 11

The willingness of saddle producers to supply saddles can be modeled by the following function: where saddles are sold for p thousand dollars. Find the producers' revenue when the market price is $7500. Round your answer to the nearest million dollars.

A) $84 million
B) $12 million
C) $3 million
D) $24 million
E) $90 million

Question 12

The willingness of saddle producers to supply saddles can be modeled by the following function: where saddles are sold for p thousand dollars. How many saddles will producers supply when the market price is $5000?

A) 6645 saddles
B) 30,000 saddles
C) 4347 saddles
D) 3323 saddles
E) 0 saddles

Question 13

Evaluate the improper integral if it converges, or state that it diverges.

A)
B)
C)
D)
E) diverges

Question 14

The average quantity of sculptures that consumers will demand can be modeled as and the average quantity that producers will supply can be modeled as where the market price is p hundred dollars per sculpture. How much are consumers willing and able to spend for 74 sculptures? Round your answer to the nearest dollar.

A) $110,433
B) $3907
C) $772,856
D) $114,340
E) $658,516

Question 15

For the 2002 fiscal year, Lowe's Companies, Inc., reported an annual net income of $1.023 billion. Assume the income can be reinvested continuously at an annual rate of return of 9.0% compounded continuously. Also assume that Lowe's will maintain this annual net income for the next 7 years. What is the future value of its 7-year net income? Round your answer to three decimal places.

A) $5.313 billion
B) $7.644 billion
C) $22.345 billion
D) $9.976 billion
E) $4.663 billion

Question 16

The average quantity of sculptures that consumers will demand can be modeled as and the average quantity that producers will supply can be modeled as where the market price is p hundred dollars per sculpture. Determine the total social gain when sculptures are sold at the equilibrium price. Round your answer to the nearest dollar.

A) $295,907
B) $132,272
C) $182,553
D) $264,371
E) $700,605

Question 17

Evaluate the improper integral if it converges, or state that it diverges.

A)
B)
C)
D)
E) diverges

Question 18

There were once more than 1 million elephants in West Africa. Now, however, the elephant population has dwindled to 19,000. Each year 17.8% of West Africa elephants die or are killed by hunters. At the same time, elephant births are decreasing by 13% per year. Considering that 47 elephants were born in the wild this year, estimate the elephant population of West Africa 27 years from now. You may use technology to calculate the answer.

A) 334 elephants
B) 111 elephants
C) 116 elephants
D) 457 elephants
E) 96 elephants

Question 19

The willingness of saddle producers to supply saddles can be modeled by the following function: where saddles are sold for p thousand dollars. At what price will the producer supply 10 thousand saddles? Round your answer to the nearest dollar.

A) $1270 per saddle
B) $55,745 per saddle
C) $8178 per saddle
D) $5174 per saddle
E) $8656 per saddle

Question 20

The daily demand for beef can be modeled by where the price for beef is p dollars per pound. Like-wise, the supply for beef can be modeled by where the price for beef is p dollars per pound. Find the point of market equilibrium. Round your answer to two decimal places.

A) $16.49 per pound, 23.65 million pounds
B) $23.65 per pound, 16.49 million pounds
C) $16.49 per pound, 1093.27 million pounds
D) $1093.27 per pound, 16.49 million pounds
E) $3.33 per pound, 1093.27 million pounds

Question 21

Find a general solution for the following differential equation:

A)
B)
C)
D)
E)

Question 22

The height h, in feet, of a certain tree increases at a rate that is inversely proportional to time t, in years. The height of the tree is 8 feet at the end of 3 years, and reaches 23 feet at the end of 6 years. Write a differential equation describing the rate of change of the height of the tree, where k is the constant of proportionality (to be determined).

A) feet per year
B) feet per year
C) feet per year
D) feet per year
E) feet per year

Question 23

The rate of change with respect to time of the quantity q of pain reliever in a person's body t hours after the individual takes the medication is proportional to the quantity of medication remaining. Assume that 5 hours after a person takes 400 milligrams of a pain reliever, one-half of the original dose remains. How much pain reliever will remain after 10 hours? Round your answer to the nearest milligram.

A) 87 milligrams
B) 76 milligrams
C) 29 milligrams
D) 25 milligrams
E) 100 milligrams

Question 24

Between 1975 and 1980, a country's energy production was increasing at a constant rate of 6.4 million Btu per year. In 1980 the country produced 37.1 million Btu. Using the initial condition, determine the particular solution for the county's energy production, where t is the number of years after 1975.

A) million Btu
B) million Btu
C) million Btu
D) million Btu
E) million Btu

Question 25

Find a general solution for the following differential equation:

A)
B)
C)
D)
E)

Question 26

Find a general solution for the following differential equation:

A)
B)
C)
D)
E)

Question 27

The height h, in feet, of a certain tree increases at a rate that is inversely proportional to time t, in years. The height of the tree is 9 feet at the end of 3 years, and reaches 22 feet at the end of 9 years. Give the particular solution for this height of the tree after t years. Round as necessary to two decimal places.

A) feet
B) feet
C) feet
D) feet
E) feet

Question 28

Between 1975 and 1980, a country's energy production was increasing at a constant rate of 6.6 million Btu per year. In 1980 the country produced 25.9 million Btu. Write a general solution for the county's energy production where t is the number of years after 1975.

A) million Btu
B) million Btu
C) million Btu
D) million Btu
E) million Btu

Question 29

The number of patents issued for a certain product between 1950 and 1970 was increasing with respect to time at a rate jointly proportional to the number of patents already obtained and to the difference between the number of patents already obtained and the carrying capacity of the system. The carrying capacity was approximately 9000 patents, and the constant of proportionality was about 0.00068. By 1951, 3000 patents had been obtained. Find a particular solution for the number of patents t years after 1950.

A) patents
B) patents
C) patents
D) patents
E) patents

Question 30

Find a general solution for the following differential equation:

A)
B)
C)
D)
E)

Question 31

The number of patents issued for a certain product between 1950 and 1970 was increasing with respect to time at a rate jointly proportional to the number of patents already obtained and to the difference between the number of patents already obtained and the carrying capacity of the system. The carrying capacity was approximately 5000 patents, and the constant of proportionality was about 0.0003. By 1960, 2000 patents had been obtained. Write a differential equation describing the rate of change in the number of patents with respect to the number of years since 1950.

A) patents per year
B) patents per year
C) patents per year
D) patents per year
E) patents per year

Question 32

The rate of change with respect to time of the quantity q of pain reliever in a person's body t hours after the individual takes the medication is proportional to the quantity of medication remaining. Assume that 2 hours after a person takes 500 milligrams of a pain reliever, one-half of the original dose remains. Find a particular solution which gives the quantity of pain reliever in a person's body after t hours. Round coefficients to two decimal places.

A) milligrams
B) milligrams
C) milligrams
D) milligrams
E) milligrams

Question 33

Assume the rate of change in the height h of a tree with respect to its age a is inversely proportional to the tree's height (with a proportionality constant of one). If the tree is 23 feet tall at age 21, how tall was the tree at age 5? Round your answer to the nearest foot.

A) 2 feet
B) 3 feet
C) 10 feet
D) 22 feet
E) 21 feet

Question 34

For the first 9 months of life, the average weight w, in pounds, of a certain breed of dog increases at a rate that is inversely proportional to time t, in months. A 1-month-old puppy's weight is changing at a rate of 8 pounds per month, and a 9-month-old puppy weighs 90 pounds. Write a differential equation describing the rate of change of the weight of the puppy.

A) pounds per month
B) pounds per month
C) pounds per month
D) pounds per month
E) pounds per month

Question 35

The number of patents issued for a certain product between 1950 and 1970 was increasing with respect to time at a rate jointly proportional to the number of patents already obtained and to the difference between the number of patents already obtained and the carrying capacity of the system. The carrying capacity was approximately 8000 patents, and the constant of proportionality was about 0.00024. By 1960, 4000 patents had been obtained. Find the general solution for the number of patents t years after 1950.

A) patents
B) patents
C) patents
D) patents
E) patents

Question 36

For the first 9 months of life, the average weight w, in pounds, of a certain breed of dog increases at a rate that is inversely proportional to time t, in months. A 1-month-old puppy's weight is changing at a rate of 7 pounds per month, and a 9-month-old puppy weighs 105 pounds. Find a particular solution for the weight of the puppy as a function of time. Round as necessary to one decimal place.

A) pounds
B) pounds
C) pounds
D) pounds
E) pounds

Question 37

Barometric pressure p (measured in inches of mercury) decreases with respect to altitude a (measured in feet) at a rate that is directly proportional to the altitude. Assume the constant of proportionality is 0.003. Write a differential equation representing the rate of change of barometric pressure.

A) inches of mercury per foot
B) inches of mercury per foot
C) inches of mercury per foot
D) inches of mercury per foot
E) inches of mercury per foot

Question 38

Find a general solution for the following differential equation:

A)
B)
C)
D)
E)

Question 39

Find a general solution for the following differential equation:

A)
B)
C)
D)
E)

Question 40

The rate of change with respect to time of the quantity q of pain reliever in a person's body t hours after the individual takes the medication is proportional to the quantity of medication remaining. Assume that 4 hours after a person takes 300 milligrams of a pain reliever, one-half of the original dose remains. Write a differential equation for the rate of change of the quantity of pain reliever in the body.

A) milligrams per hour
B) milligrams per hour
C) milligrams per hour
D) milligrams per hour
E) milligrams per hour

Question 41

A person learns a new task at a rate that is equal to the percentage of the task not yet learned. Let p represent the percentage of the task already learned at time t (in hours). If a person begins learning at a rate of 19 percent per hour, write a differential equation describing the rate of change in the percentage of the task learned at time t.

A) percent per hour
B) percent per hour
C) percent per hour
D) percent per hour
E) percent per hour

Question 42

Find a general solution for the following differential equation:

A)
B)
C)
D)
E)

Question 43

The rate of change in the number of jobs J for a Michigan roofing company is increasing by approximately 7.2 jobs per month squared. The number of jobs in January (m = 1) is decreasing at the rate of 1.7 jobs per month, and company records indicate that the company had 72 roofing jobs in February. Write a differential equation for the rate at which the rate of change in the number of roofing jobs for this company is changing.

A) jobs per month
B) jobs per month
C) jobs per month squared
D) jobs per month squared
E) jobs per month

Question 44

The rate of change in the number of jobs J for a Michigan roofing company is increasing by approximately 8 jobs per month squared. The number of jobs in January (m = 1) is decreasing at the rate of 1.7 jobs per month, and company records indicate that the company had 86 roofing jobs in February. Estimate the number of roofing jobs in August.

A) 326 jobs
B) 268 jobs
C) 392 jobs
D) 296 jobs
E) 115 jobs

Question 45

Newton's Law of Cooling says that the rate of change (with respect to time t in minutes) of the temperature T of an object is proportional to the difference between the temperature of the object and the temperature A of the object's surroundings. Initially, the object has a temperature of If the object is placed in a room where the temperature is fixed at and the object begins to cool at a rate of 1.5 degrees per minute, use Euler's method (with one step) to estimate the temperature of the object after 15 minutes. Round your answer to one decimal place.

A)
B)
C)
D)
E)

Question 46

Newton's Law of Cooling says that the rate of change (with respect to time t in minutes) of the temperature T of an object is proportional to the difference between the temperature of the object and the temperature A of the object's surroundings. Initially, the object has a temperature of If the object is placed in a room where the temperature is fixed at and the object begins to cool at a rate of 2.3 degrees per minute, determine the constant of proportionality. Round your answer to four decimal places.

A) -0.3333
B) -2.3000
C) -0.1667
D) -4.6000
E) -6.0000

Question 47

Newton's Law of Cooling says that the rate of change (with respect to time t in minutes) of the temperature T of an object is proportional to the difference between the temperature of the object and the temperature A of the object's surroundings. If the object is placed in a room where the temperature is fixed at 71.4 degrees Fahrenheit, write a differential equation describing this law.

A) degrees Fahrenheit per minute
B) degrees Fahrenheit per minute
C) degrees Fahrenheit per minute
D) degrees Fahrenheit per minute
E) degrees Fahrenheit per minute

Question 48

Between 1919 and 1995, the rate of change in the rate of change of the postage required to mail a first-class 1-ounce letter was approximately 0.022 cent per year squared. The postage was 2 cents in 1919, and it was increasing at the rate of approximately 0.393 cent per year in 1958. Using this information, estimate the postage that will be required to mail a first-class, 1-ounce letter in 2088.

A) $3.46
B) $3.26
C) $2.38
D) $1.46
E) $1.56

Question 49

A person learns a new task at a rate that is proportional to the percentage of the task not yet learned. Let p represent the percentage of the task already learned at time t (in hours). If a person begins learning at a rate of 39 percent per hour, estimate the percentage of the task that will be learned in 2 hours using Euler's method with a step size of 0.5. Round your answer to one decimal place.

A) 58.0 %
B) 66.8 %
C) 64.8 %
D) 49.4 %
E) 66.3 %

Question 50

When a spring is stretched and then released, it oscillates according to two laws of physics: Hooke's Law and Netwon's Second Law. These two laws combine to form the following differential equation in the case of free, undamped oscillation: where m is the mass of an object attached to the spring, x is the distance the spring is stretched beyond its standard length with the object attached (its equilibrium point), t is time, and k is a constant associated with the strength of the spring. Consider a spring with from which is hung a 40-pound weight. The spring with the weight attached stretches to its equilibrium point. The spring is then pulled 7 feet farther than its equilibrium and released. Find a particular solution for the position of the spring after time t. Use the fact that , where g is the gravitational constant 32 feet per second per second, and that when the spring is first released, its velocity is zero. Round the coefficients to three decimal places.

A)
B)
C)
D)
E)

Question 51

When a spring is stretched and then released, it oscillates according to two laws of physics: Hooke's Law and Netwon's Second Law. These two laws combine to form the following differential equation in the case of free, undamped oscillation: where m is the mass of an object attached to the spring, x is the distance the spring is stretched beyond its standard length with the object attached (its equilibrium point), t is time, and k is a constant associated with the strength of the spring. Consider a spring with from which is hung a 68-pound weight. The spring with the weight attached stretches to its equilibrium point. The spring is then pulled 3 feet farther than its equilibrium and released. Write a differential equation describing the acceleration of the spring with respect to time t measured in seconds. Use the fact that , where g is the gravitational constant 32 feet per second per second.

A) feet per second per second
B) feet per second per second
C) feet per second per second
D) feet per second per second
E) feet per second per second