A fast-food restaurant wants a special container to hold coffee. The restaurant wishes the container to quickly cool the coffee from 200 degrees to 130 degrees F and keep the liquid between 110 degrees and 130 degrees F as long as possible. The restaurant has three containers to select from.

- The CentiKeeper Company has a container that reduces the temperature of a liquid from 200 degrees F to 100 degrees F in 30 minutes by maintaining a constant temperature of 70 degrees F.
- The TempControl Company has a container that reduces the temperature of a liquid from 200 degrees F to 110 degrees F in 25 minutes by maintaining a constant temperature of 60 degrees F.
- The Hot’n’Cold Company has a container that reduces the temperature of a liquid from 200 degrees F to 120 degrees F in 20 minutes by maintaining a constant temperature of 65 degrees F.

You need to recommend which container the restaurant should purchase.

(a) Use Newton’s Law of Cooling to find a function relating the temperature of the liquid over time for each container.

(b) How long does it take each container to lower the coffee temperature from 200 degrees to 130 degrees F?

(c) How long will the coffee temperature remain between 110 degrees and 130 degrees F? This temperature is considered the optimal drinking temperature.

(d) Graph each function using a graphing utility.

(e) Which company would you recommend to the restaurant? Why?

(f) How might the cost of the container affect your decision?

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