A can of soft drink is initially at a temperature of 72.3◦F. Let t be the number of minutes since the can is placed into a fridge, set at 37.7◦F. The temperature of the drink, 90 minutes after being placed in the fridge, is 59.5◦F.

Question

Suppose the formula for the temperature T , in ◦F, is T = ae−k t +c where a, k, and c are constants.

(a) Determine the units, if possible, of the constants a and k.

(b) Use algebraic reasoning, and your understanding of the physical situation to determine the exact values of a, k, and c. Explain your reasons.

(c) The formula we have ceases to be useful when the temperature is less than 1◦F away from the temperature of the fridge. What time does this occur?

(d) Calculate the rate of change of the drink over the interval 25 É t É 30. What are the units? What is the meaning of this number?

(e) If the temperature of the can of drink started at 65◦F but the other data in the question stayed the same, would the value of k be larger or smaller?

Answer 1

(a) well you have ae^-kt is °F. And you know that t is in °F, so a is just a number if e^-k(°F) is also °F. so k is -1/°F

we know that as t→∞, f(t) → 37.7 so c=37.7

(b) using the given data, we have the points (0,72.3) and (90,59.5)
a + 37.7 = 72.3
so a = 34.6
Now we have
34.6 e^(-90k) + 37.7 = 59.5
k = 0.00513

I expect you can answer the rest of the questions, ok?

Answer 2

Thank you so much, why is a + 37.7 = 72.3 instead of the initial temp?

I also don't know how to attempt c and d