calc - math Question

Question

You pour yourself a cup of coffee. The function that models the cooling of 73◦C degree cup of coffee in a room of temperature 20◦C is given by T(t) = 20 + 53e^kt .

where the temperature T is (C), time t is in minutes (min), and k < 0 is an unknown constant.

Suppose that 5 minutes later, the cup of coffee is at a temperature of 65◦C.

(a) Find the unknown constant k.

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(b) How long will it take to for the cup of coffee to cool to 45◦C? Be accurate to the minute.

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(c) How fast is the coffee cooling 10 minutes after pouring? Be accurate to two decimal places

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(d) How many minutes after pouring will the coffee cooling at a rate of 1◦C/min? Be accurate to the minute

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Answer 1

To find the value of k, let T = 65 and t = 5, then solve for k. Isolate the term e^5k, then take natural log of both sides, dividing both sides by 5 to get the value of k.

Now that you have the function defined, let T = 45 and solve for t. This is done by isolating the term e^kt, natural log of both sides, etc.

The last two parts are dealing with rate of change, i.e. derivative. T'(t) = 53ke^kt. When t = 10, what is T'(10)?

To determine at what time the derivative will be about -1, set T'(t) = -1 and solve for t.

Use the following to confirm your answers after you do the work.

desmos.com/calculator/tyrtlye6ov