Question

A body of unknown temperature is placed in a refrigerator at a constant temperature of 0∘F, If after 20 minutes the temperature of the body is 40∘F, and after 40 minutes the temperature of the body is 20∘F, find the initial temperature of the body.

Answer 1

$\text{The change in temperature is given by }\\ \frac{d\theta}{dt}= k(0-\theta)\\ \implies \frac{d\theta}{-\theta}=kdt\\ \implies \theta = 17-ce^{-kt}-(1)\\ \text{at t = 60 seconds and $\theta = 20$, we have that}\\ -3=ce^{-60k}-(3)\\ \text{at t = 30seconds and $\theta = 27$, we have that}\\ \text{Dividing (3) by (2), we have}\\ 0.3 = e^{-30k}\\ \implies k = 0.0401\\ \text{Putting k =0.0401 in 3 we have that}\\ c = -33.33\\ \text{Next, we substitute k and c in (1), at time 0 to get our solution}\\ \therefore \theta = 17+33.33e^0=50.33^0C$