Exam 6: Analyzing Accumulated Change: Integrals in Action

Find a general solution for the following differential equation:

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?

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.

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).

Find a general solution for the following differential equation:

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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 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.

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.