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HOT COFFEE:

Your breakfast restaurant needs to buy a container to holdcoffee. Generally, restaurants wish to have container that caneither quickly cool the coffee to a drinkable range, and/or keepthe coffee in that drinkable range for as long as possible. Quiteoften, one container is not the “best” at achieving both of theseoutcomes.

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

Project Information:

The drinkable range for coffee is considered to be between 130°Fto 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 of70°F.

TempControl This container reduces the temperature of the liquid from 200°to 105°F in 20 minutes by maintaining a constant temperature of60°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 of65°F.

Calculations

ï‚· Your math work does not need to be typed, that can be writtenby hand, but it should be done neatly.

ï‚· Graphs MUST be printed out using TI-Connect software for theTI calculator or some other graphing software. (This is availablein the Math Centers)

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

For each of these containers, provide thesecalculations:

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

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

ï‚· u(t) is your temperature after time t, inminutes;

ï‚· T is the temperature of the surrounding medium (as inair temperature in

the room);

ï‚· u0 indicates the initial temperature;

ï‚· k is the constant (which in this case should benegative, since the

temperature is cooling

 For finding k, use and initial temperature of200°F.

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

each model of container.

Next, figure out how long it takes the containers to cool thecoffee 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 ofthe drinkable range

(110o).

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

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

ï‚· 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 particularcontainer – citing your math facts. Include a summary table. (seeinstructions above)

Summary Page

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

ï‚· Create a table summarizing the results of your calculations.For each container include the time it takes for the coffee toreach a drinkable temperature and how long the coffee remains inthe drinkable range.

ï‚· Based on your calculations choose the model which will be bestsuited for your restaurant. Write a short paragraph explaining indetail why you believe the model you chose is the best model foryour restaurant keeping in mind your customer base.