Projects - Cengage Learning

Appendix C C-37
PROJECT
Coffee Temperature
Suppose that a cup of hot coffee at a temperature of T 0 is set down to cool in a
room where the temperature is kept at T 1 . Then the temperature of the coffee as
it cools can be modeled by the function
f(t) (T 0 T 1 )e kt T 1
(1)
Here, f(t) is the temperature of the coffee after t minutes; t 0 corresponds to
the initial instant when the temperature of the hot coffee is T 0 ; and k is a (negative)
constant that depends, among other factors, on the dimensions of the cup
and the material from which it is constructed. [This model is derived in calculus.
It is based on Newton’s law of cooling: The rate of change of temperature
of a cooling object is proportional to the difference between the temperature of
the object and the surrounding temperature. (Note: y is proportional to x means
y kx for some constant k.)]
After studying the following example, solve Problem A below by applying
equation (1) and using the technique shown in the example. Then try your hand
at Problem B. You won’t be able to complete Problem B using the techniques
developed in this text up to now. In complete sentences, explain exactly at
which point you get stuck. What would you have to know how to do to complete
the solution? (In the next section we discuss logarithms, which will allow
us to complete Problem B.) For now, use a graphing utility to obtain an approximate
answer for Problem B.
EXAMPLE
Coffee Temperature Project
Given that the graph of the function y ae kx passes through the points (0, 2)
and (3, 5), find y when x 7.
SOLUTION
Substituting x 0 and y 2 in the given equation yields 2 ae 0 , and therefore
a 2. Next, substituting x 3 and y 5 in the equation y 2e kx yields
5 2e k(3) , or e 3k 2.5. This last equation can be rewritten (e k ) 3 2.5. Taking
the cube root of both sides gives us
e k 2.5 13
(2)
The original function can now be written as follows
y 2e kx 2(e k ) x
2(2.5 13 ) x
using equation (2)