Chapter 10.4, Problem 44PS

Solution

a.

To state: The function $P\left(t\right)$ for the power used by the microwave if P is the power (in watts) used by a microwave oven is the product of the voltage V (in volts) and the current I (in amperes). The voltage and current can be modeled by: $V=170\cos 120\pi t\text{ and }I=11.3\cos 120\pi t$

The resultant answer is $P\left(t\right)=1921{\cos }^{2}120\pi t$ .

Given information:

The given models of the voltage and current are $V=170\cos 120\pi t\text{ and }I=11.3\cos 120\pi t$ .

Explanation:

Consider the given models:

  $V=170\cos 120\pi t\text{ and }I=11.3\cos 120\pi t$

Here t represents the time in seconds.

The power is the product of the voltage V (in volts) and the current I (in amperes).

  $P=V\cdot I$

Substitute the values $V=170\cos 120\pi t\text{ and }I=11.3\cos 120\pi t$ ,

  $\begin{array}{c}P\left(t\right)=\left(170\cos 120\pi t\right)\left(11.3\cos 120\pi t\right)\\ P\left(t\right)=1921{\cos }^{2}120\pi t\end{array}$

Therefore, the function is $P\left(t\right)=1921{\cos }^{2}120\pi t$ .

b.

To state: The time at which the microwave uses the 375 watts of power.

The resultant answer is $t=\frac{1}{120\pi }{\cos }^{-1}\left(\sqrt{\frac{375}{1921}}\right)$ .

Given information:

The given power is 375 watts.

Explanation:

Consider the function from part(a) $P=1921{\cos }^{2}120\pi t$ ,

  $P=1921{\cos }^{2}120\pi t$

Now substitute the values $P=375\text{ watts}$ and solve for t ,

  $\begin{array}{c}375=1921{\cos }^{2}120\pi t\\ \frac{375}{1921}={\cos }^{2}120\pi t\\ \sqrt{\frac{375}{1921}}=\cos 120\pi t\\ 120\pi t={\cos }^{-1}\left(\sqrt{\frac{375}{1921}}\right)\\ t=\frac{1}{120\pi }{\cos }^{-1}\left(\sqrt{\frac{375}{1921}}\right)\end{array}$

Therefore, the value is $t=\frac{1}{120\pi }{\cos }^{-1}\left(\sqrt{\frac{375}{1921}}\right)$ .

c.

To graph: The function $P\left(t\right)$ . Also describe the difference of the graph from the cosine function of the form $y=a\cos bt$ .

Given information:

From part (a) the function is $P\left(t\right)=1921{\cos }^{2}120\pi t$ .

Graph:

Consider the function $P\left(t\right)=1921{\cos }^{2}120\pi t$ ,

Graph the function $P\left(t\right)=1921{\cos }^{2}120\pi t$

  $P=1921{\cos }^{2}120\pi t$

using a graphing calculator:

  

The given cosine function is $y=a\cos bt$ . Now compare the function $P\left(t\right)=1921{\cos }^{2}120\pi t$ with $y=a\cos bt$ .

The graph of $P\left(t\right)=1921{\cos }^{2}120\pi t$ is a vertically shift of graph of $y=a\cos bt$ .

Interpretation: The difference between the graph of $P\left(t\right)$ and the graph of the function $y=a\cos bt$ is that the graph of $P\left(t\right)$ is vertically shifted.