Cola Consumption

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/