Question

g When a cake is removed from an oven and placed on a kitchen counter, its temperature is measured at 300◦F. Three minutes later its temperature is 250◦F. Write the initial value problem describing the change in the temperature of the cake, if the temperature of the kitchen is 70◦F.

Answer 1

`T = 70 +230 ((18)/(23))^{(t)/(3)}`

Step-by-step explanation:

We use T to represent the temperature of the cake at time "t"

Also
`T_k` to be the kitchen temperature :

By differentiation:

`(dT)/(dt)=-k (T-T_k)\\ \\(dT)/(dt)=-k (T-70)\\\\`

where;

T(0) = 300° F

T(3) = 250° F

`T = 70 + Ce^(-kt)`

`300 = 70 + Ce^0\\\\300-70 = C*1\\\\C = 230\\\\T = 70 + 230 e^(-kt)`

T(3) = 250

`250 = 70 + 230 e^(-3k)`

`250 - 70 = 230 e^(-3k)\\\\180 = 230 e^(-3k)\\\\(180)/(230)= e^(-3k)\\\\(18)/(23)= e^(-3k)\\\\-k = (1)/(3)In((18)/(23))\\\\T = 70 + 230 e^{(t)/(3)In (18)/(23)}\\\\T = 70 +230 ((18)/(23))^{(t)/(3)}`