Exam 6: Analyzing Accumulated Change: Integrals in Action

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.

Find a general solution for the following differential equation:

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.

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.

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.

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 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.

Find a general solution for the following differential equation:

Find a general solution for the following differential equation:

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.

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.

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.

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.

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

Find a general solution for the following differential equation:

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.

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?

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.

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.

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.