Project 1: Exponential and Logarithmic Functions

Hot Coffee: A fast-food restaurant wants a special container to hold coffee from 200° to 130°F and keep the liquid between 110° and 130°F as long as possible. The restaurant has three containers to select from. Recommend which container the restaurant should purchase.

The CentiKeeper Company has a container that reduces the temperature of a liquid from 200° 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° to 110°F in 25 minutes by maintaining a constant temperature of 60°F.

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

Use Newton's Law of Cooling to find a function relating the temperature of the liquid over time for each container:

Function: U(t)= T+(U0-T)e^kt
CentiKeeper Company: U0=200, T=70, t=30 U=100

Temp Control Company: U0=200, T=60, t=25, U=110

Hot'n'Cold Company: U0=200, T=65, t=20, U=120

How long does it take each container to lower the coffee temperature from 200° to 130°F?

CentiKeeper Company: t= ln(60/130)/(-0.048877902) = 15.81880min

Temp Control Company: t= ln(70/140)/(-0.041184777) = 16.83018min

Hot'n'Cold Company: t= ln(65/135)/(-0.041184777) = 16.27918min

How long will the coffee temperature remain between 110° and 130°F? This temperature is considered the optimal drinking temperature.

What we want to do is find the time it takes to reach 110°F then take the difference of that and the results from above.
CentiKeeper Company: t = ln(40/130)/(-0.048877902) = 24.11427min

Temp Control Company: t = ln(50/140)/(-0.041184777) = 25min

Hot'n'Cold Company: t= ln(45/135)/(-0.041184777) = 24.46957min

Time the temperature will be between 110 and 130.
CentiKeeper Company: t = 24.11427min - 15.81880min = 8.29547min

Temp Control Company: t = 25min - 16.83018min = 8.16982min

Hot'n'Cold Company: t = 24.46957min - 16.27918min = 8.19039min

Graph each function using a graphing utility:

CentiKeeper Company = Red

Temp Control Company = Purple

Hot'n'Cold Company = Blue

What company would you recommend to the restaurant? Why?

I would recommend the CentiKeeper Company because compared to the other companies it cools down the faster while maintaining its temperature the longest.

How might the cost of the container affect your decision?

The cost of the container greatly affects the decision to go with one company over the other because there are only slight differences between the cups. All three companies cool down the contents of the cup in less than 17 minutes and all three of them hold the ideal temperature for more than 8 minutes. When it comes to recommending a company over another if I had the cost of each cup I would have recommended the cheapest one.

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