Name
LESSON
7-5
Date
Class
Challenge
Exponential Heating and Cooling
Newton’s Law of Cooling states that the rate of heat loss
of an object is proportional to the difference in temperatures
between the object and its surrounding ambient temperature.
T t TA
This phenomenon is modeled with a differential equation
t TA
T
and that equation may be solved to give
_________
T t ⫽ T A ⫹ [ T 0 ⫺ T A] b t
where T t is the varying temperature of the object at a
given time, t, T A is the surrounding ambient temperature,
T 0 is the initial temperature of the object, and b is a
constant that depends on the material the object is
composed of and how fast it heats or cools.
Suppose you decided to make a cup of hot chocolate
heated to 180°F in the kitchen that is at 72°F.
1. Solve the above equation for the constant b.
2. If the cup of hot chocolate cooled to 150°F in
15 minutes, find the value of the constant b in
the above equation. Express your answer to
five decimal places.
3. Solve the above equation for t.
4. Suppose you like your hot chocolate at the tepid
temperature of 120°F. How long, to the nearest
minute, will you have to wait until it cools to this
temperature?
T 0 T A b t
bt
T0 TA
T t TA
t log b
log _________
T0 TA
T t TA
log _________
T0 TA
______________
log b
t
10
log
TT t TT A
_________
0
A
______________
t
b
b ; 0.97854
First 3 steps same as #1;
t TA
T
__________
T0 TA
log ____________
t
log b
About 37 min
To go along with your hot chocolate, you take a frozen cherry pie from the
freezer and place it in the oven preheated to 350°F. Assume the freezer is
at 32°F.
5. If the cherry pie comes to a temperature of 120°F in
20 minutes, find the value of the constant b in the
above equation. Express your answer to 5 decimal
places.
b ; 0.98362
6. How long will it take for the pie to reach its final
temperature of 220°F?
About 55 min
7. The pie is taken out of the oven and set on a table
in a room at 80°F. In 10 minutes it has cooled to
185°F. However, the pie must cool to 125°F
before it is ready to eat. How much longer will
you have to wait?
About 30 min
Copyright © by Holt, Rinehart and Winston.
All rights reserved.
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