Chapter 8.5, Problem 61E

Cost The temperature T (in degrees Fahrenheit) in a house is given by

$T=72+12\sin \frac{\pi \left(t-8\right)}{12}$ where t is the time (in hours), with $t=0$ corresponding to midnight. The hourly cost of cooling a house is $0.30 per degree.

Find the cost C of cooling this house between 8 A.M. and 8 P.M. when the thermostat is set at 72°F (see figure) by evaluating the integral

$C=0.3{\displaystyle {\int }_8^{20}\left[72+12\sin \frac{\pi \left(t-8\right)}{12}-72\right]dt.}$

(b) Find the saving realized by resetting the thermostat to 78°F (see figure) by evaluating the integral

$C=0.3{\displaystyle {\int }_{10}^{18}\left[72+12\sin \frac{\pi \left(t-8\right)}{12}-78\right]dt.}$