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# Newton's Law of Cooling: Which container should be purchased?

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

(1) 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.
(2) 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 60 F.
(3) The Hot'n'Cold Company has a container that reduces the temperature of a liquid from 200 F to 120 F in 20 minutes by maintaining a constant temperature of 65 F.

You need to recommend the container the restaurant should purchase.

(a) Use Newton's Law of Cooling to find a function relating the temperature of a liquid over time.
(b) Graph each function using a graphing utility.
(c) How long does it take each container to cool the coffee temperature from 200 F to 130 F?
(d) How long will coffee temperature be between 110 F and 130 F? This temperature is considered the optimum drinking temperature.
(e) Which company would you recommend to the restaurant? Why?
(f) How might the cost of the container affect your decision?