Exam 6: Analyzing Accumulated Change: Integrals in Action

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.

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?

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.

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

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.

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?

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.

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

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?

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.

Find a general solution for the following differential equation: