Date: 05/03/99 at 19:39:11 From: Kim Pirrone Subject: AP Calculus Review The rate of consumption of Cola in the U.S. is given be S(t)= Ce^kt where S is measured in billions of gallons per year and t is measured in years from the beginning of 1980. a. The consumption rate doubles every 5 years and the consumption rate at the beginning of 1980 was 6 billion gallons per year. Find C and k. b. Find the average rate of consumption of cola over the 10 year time period beginning Jan. 1, 1983. Indicates units of measure. c. Use the trapezoidal rule with four equal subdivisions to estimate the integral S(t) dt from 5 to 7. d. Using correct units, explain the meaning of integral S(t) dt from 5 to 7 in terms of cola consumption.
Date: 05/03/99 at 21:38:46 From: Doctor Jaffee Subject: Re: AP Calculus Review Hi Kim, I recognize this problem. It was on the AP exam a few years ago. The Calculus students at my school have been working on this same problem. Here is what they suggest. In 1980, t = 0, so S(0) = 6 and S(5) = 12. From these two facts you should be able to determine C, then k. Hint: S(0) = 6 = Ce^k(0) and S(5) = 12 = Ce^k(5). Use the formula for "the average of a function" to solve part b. You should be able to find it in the discussion of the Mean Value Theorem for Integrals. Part c is pretty straightforward. Make sure you round your answer according to the specifications of the AP exam. In part d keep in mind that the curve tells you the rate of consumption measured in gallons per year and the horizontal axis is measured in years since 1980. Since the definite integral is just a way of multiplying the length of the inteval of the horizontal axis by the values of y (which may be varying), the integral measures billions of gallons per year times the number of years. I hope this helps a little. Good luck with finishing this problem; write back if it needs more explanation or clarification, and, in particular, good luck on the exam. - Doctor Jaffee, The Math Forum http://mathforum.org/dr.math/
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