Download Report

Name: ____________________________________________ Date: ________________________ Class: ___________
Newton’s Law of Cooling Pre-Calculus Project
Due Date: Friday, May 1, 2015
A fast-food restaurant wants a special container to hold coffee. The restaurant wishes the container to quickly cool the
coffee from 200°F to 130°F and keep the liquid between 110°F and 130°F for as long as possible. The restaurant has
3 containers to select from:
The CentiKeeper Company has a container that reduces the temperature of a liquid from 200°F to 100°F in
30 minutes by maintaining a constant temperature of 70°F.
The TempControl Company has a container that reduces the temperature of a liquid from 200°F to 110°F in
25 minutes by maintaining a constant temperature of 68°F.
The Hot-n-Cold Company has a container that reduces the temperature of a liquid from 200°F to 120°F in
minutes by maintaining a constant temperature of 72°F.
20
You need to recommend which container the restaurant should purchase.
a) Use Newton’s Law of Cooling to find the functions, C(t), T(t), and H(t), respectively, for each of the containers
above relating the temperature of the liquid over time. (That is, the independent variable is time in minutes and
the dependent variable is temperature in degrees Fahrenheit). Within the equations, the k-values found must be
rounded to the ten-thousandths place.
b) How long does it take each container to lower the coffee temperature from 200°F to 130°F? Answers should be
rounded to the nearest hundredth.
c) How long will the coffee temperature in each container remain between 110°F and 130°F? This temperature is
considered the optimal drinking temperature. Answers should be rounded to the nearest hundredth.
d) Make a full-page graph with all three functions on a single set of axes. Color code each graph, and include a
legend to identify each graph by color. To clearly show each cup’s temperature changes from 130°F to 110°F,
use a domain (calculator window) to x [15, 26] and y [105,135] to clearly show each cup’s temperature
changes from 130°F to 110°F.
e) Disregarding cost of the cups, which company would you recommend to the restaurant? Why?
f) A Marketing company, through extensive research and polling, has determined that for each 30-second loss of
time that the coffee is hot (i.e. its temperature is between 130°F and 110°F), sales decrease by 150 units. For
example, if the restaurant sells 4,000 cups of coffee in cups that keep the coffee hot for 12 minutes, then change
the coffee container to a cup that only keeps the coffee hot for 9 minutes, sales would decrease by 900 units, so
they’d only sell 3,100 cups of coffee in the same time period.
i) Use the data in the example above to write a linear function, S(t), that describes the number of
expected sales in one month as a function of the time t that the coffee stays hot. Hint: the statement
beginning with “For example” provides to coordinate points. Use these points to find slope and yintercept.
ii) Use your answers from part (c) to find the number of expected sales for the CentiKeeper,
TempControl, and Hot-n-Cold cups.
iii) Suppose CentiKeeper charges a base fee of $75 and $29.95 per 1000 cups. TempControl charges a
base fee of $60 and $32.00 per 1000 cups, and Hot-n-Cold charges a base fee of $35 and $44.00 per
1000 cups. Would cost consideration cause you to change your recommendation from part (e) above?
Justify your conclusion with a specific cost argument. Hint: You may not purchase a portion of a bag
of cups.