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### 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.

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/
```
Associated Topics:
High School Calculus

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