Chapter 8.1, Problem 22E

To determine

To calculate: the average daily consumption of electricity at the home, measured in kilowatt-hours.

Answer to Problem 22E

The average daily consumption of electricity at the home is approximately 93.6 kilowatt-hours.

Explanation of Solution

Given information:

The rate of consumption of electricity is modeled by the function $C\left(t\right)=3.9-2.4\sin \left(\frac{\pi t}{12}\right)$

Calculation:

Seek the cumulative effect of the consumption rate for $0\le t\le 24$

The amount consumed during this interval is approximately

  $\begin{array}{l}{\displaystyle \sum C\left(t_k\right)\Delta t\text{ kilowatt hours}}\\ {\displaystyle \underset{0}{\overset{24}{\int }}C\left(t\right)dt=}{\displaystyle \underset{0}{\overset{24}{\int }}\left(3.9-2.4\sin \left(\frac{\pi t}{12}\right)\right)}dt\text{ kilowatt hours}\\ {\displaystyle \underset{0}{\overset{24}{\int }}C\left(t\right)dt=}3.9t{]}_0^{24}\text{+2}\text{.4}\cdot \frac{12}{\pi }\text{cos}\left(\frac{\pi t}{12}\right){\text{]}}_0^{24}\text{ kilowatt hours}\\ {\displaystyle \underset{0}{\overset{24}{\int }}C\left(t\right)dt=}\text{93}\text{.6+0 kilowatt hours}\\ {\displaystyle \underset{0}{\overset{24}{\int }}C\left(t\right)dt\approx }\text{93}\text{.6 kilowatt hour}s\end{array}$

Therefore,

The average daily consumption of electricity at the home is approximately 93.6 kilowatt-hours.