Answer
a) $137° F$
b) $\approx$ $116$ min
Work Step by Step
a)
$\frac{dT}{dt}$ = $k(T-T_s)$
$\frac{dT}{dt}$ = $k(T-75)$
Let
$y$ = $T-75$
$y(0)$ = $T(0)-75$ = $185-75$ = $110$
$y(t)$ = $y(0)e^{kt}$
$y(t)$ = $110e^{kt}$
$y(30)$ = $110e^{30k}$
$150-75$ = $110e^{30k}$
$k$ = $\frac{1}{30}\ln{\frac{15}{22}}$
$y(t)$ = $110e^{\frac{t}{30}\ln{\frac{15}{22}}}$
$y(45)$ = $110e^{\frac{45}{30}\ln{\frac{15}{22}}}$ $\approx$ $137° F$
b)
$T(t)$ = $100$ then
$y(t)$ = $100-75$ = $25$
$y(t)$ = $110e^{\frac{t}{30}\ln{\frac{15}{22}}}$
$25$ = $110e^{\frac{t}{30}\ln{\frac{15}{22}}}$
$t$ $\approx$ $116$ min