Question

HOT COFFEE:

Your breakfast restaurant needs to buy a container to hold coffee. Generally, restaurants wish to have container that can either quickly cool the coffee to a drinkable range, and/or keep the coffee in that drinkable range for as long as possible. Quite often, one container is not the “best” at achieving both of these outcomes.

From the three container models provided, recommend the container that provides the best solution for your restaurant. (Think about situations where you may want the coffee to cool to the drinkable range sooner or situations where you would want the coffee to stay in the drinkable range longer)

Project Information:

The drinkable range for coffee is considered to be between 130°F to 110°F. Coffee is brewed at 212°F.

Here are the three models of containers:

CentiKeeper This container reduces the temperature of the liquid from 200° to 100°F in 35 minutes by maintaining a constant temperature of 70°F.

TempControl This container reduces the temperature of the liquid from 200° to 105°F in 20 minutes by maintaining a constant temperature of 60°F.

Hot’n’Cold This container reduces the temperature of the liquid from 200° to 120°F in 20 minutes by maintaining a constant temperature of 65°F.

Calculations

 Your math work does not need to be typed, that can be written by hand, but it should be done neatly.

 Graphs MUST be printed out using TI-Connect software for the TI calculator or some other graphing software. (This is available in the Math Centers)

 If you solve graphically, then you’ll need to include additional graphs showing how you arrived at the solution

For each of these containers, provide these calculations:

First, you need to find the k value for EACH container. (three separate problems)

Use Newton’s law of cooling: u(t)=T +(u0 -T)e^kt

 u(t) is your temperature after time t, in minutes;

 T is the temperature of the surrounding medium (as in air temperature in

the room);

 u0 indicates the initial temperature;

 k is the constant (which in this case should be negative, since the

temperature is cooling

 For finding k, use and initial temperature of 200°F.

Use the value k to set up the particular variation of the cooling function, u(t), for

each model of container.

Next, figure out how long it takes the containers to cool the coffee to the upper

level of the “drinkable” range (130o) – (For this and the next, use an initial temp of

212°F.)

Then, figure out the time it takes to get to the lowest level of the drinkable range

(110o).

Please show a graph for each container. The graph should contain a function for

the container and show the intersections with 130 o and 110 o temperatures in an appropriate window.

 There should be entries in Y1= u(t) , Y2=130, and Y3=110.

MAT 195 Precalculus Eng/Sci Newton’s Law of Cooling Project

6. Please type a paragraph explaining why you chose a particular container – citing your math facts. Include a summary table. (see instructions above)

Summary Page

 Write a short paragraph describing the customer base at your coffee shop and more importantly their coffee drinking habits. (Are they retirees, students, commuters? Are they lingering in the shop and getting frequent refills or taking it to go? The way in which they drink their coffee should inform your choice for a container.)

 Create a table summarizing the results of your calculations. For each container include the time it takes for the coffee to reach a drinkable temperature and how long the coffee remains in the drinkable range.

 Based on your calculations choose the model which will be best suited for your restaurant. Write a short paragraph explaining in detail why you believe the model you chose is the best model for your restaurant keeping in mind your customer base.

Answer 1