A casserole baked at 375 degrees is taken out of the oven into a kitchen that is 74 degrees. After 6 minutes, the temperature in the casserole is 320.9 degrees. After how many minutes will...

Question

A casserole baked at 375 is taken out of the oven into a kitchen that is 74 degrees. After 6 minutes, the temperature in the casserole is 320.9 degrees. After how many minutes will it's temperature be 240 degrees?

Answer 1

Let T(t) = temperature of the casserole t minutes after removal from the oven

Using Newton's Law of Cooling, T(t) = 74 + Ce^kt

Since T(0) = 375, 74 + Ce⁰ = 375 . So, C = 375 - 74 = 301

T(t) = 74 + 301e^kt

T(6) = 74 + 301e^6k = 320.9

301e^6k = 246.9

e^6k = 0.82026578

6k = ln(0.82026578)

k = -0.033

So, T(t) = 74 + 301e^-0.033t

Therefore, if T(t) = 240. then 74 + 301e^-0.033t = 240

301e^-0.033t = 166

e^-0.033t = 0.551495

-0.033t = ln(0.551495)

t = 18.034 minutes