Answer
$T \approx 125.56$ $Farenheit$
Work Step by Step
$$T = C + (T_{0} - C)e^{-kt}$$ To answer the exercise we must first find the value of $k$. To do this, we can use the information given regarding the cake's temperature after 30 minutes: $$(140) = (70) + [210 - 70]e^{-k(30)}$$ $$140 - 70 = 140e^{-30k}$$ $$\frac{1}{2} = 2^{-1} = e^{-30k}$$ $$-30k = \ln 2^{-1} = -\ln 2$$ $$k = \frac{\ln 2}{30}$$ Having the value of $k$, we can now proceed to find the value of $T$ when $t = 40$: $$T = 70 + (210 - 70)e^{-\frac{\ln 2}{30}(40)}$$ $$T = 70 + 140(e^{\ln 2})^{-\frac{40}{30}}$$ $$T = 70 + 140(2^{-\frac{4}{3}}) \approx 125.56$$
