Chapter 8.5, Problem 51AYU

(a)

To determine

To find: The graph for the function to relate $V$ and $t$ .

Answer to Problem 51AYU

The graph to relate the function to relate $V$ and $t$ .

Explanation of Solution

Given:

The expression for $V=e^{\frac{-t}{3}}\cos \left(\pi t\right)$ and the range of $t$ is $0\le t\le 3$ .

The given diagram is shown in Figure 1

  

Figure 1

Calculation:

Consider the given function is,

  $V=e^{\frac{-t}{3}}\cos \left(\pi t\right)$

By the use of MAPLE the graph of the function is shown in Figure 1

  

Figure 1

(b)

To determine

To find: The time at which $t$ does the graph of $V$ touch the graph of $y=e^{\frac{-t}{3}}$ . Also, determine the time at which the graph of $V$ touch the graph of $y=-e^{\frac{-t}{3}}$ .

Answer to Problem 51AYU

The graph for $V$ and $y=e^{\frac{-t}{3}}$ meet at $t=0,2$ and the graph for $V$ and $y=-e^{\frac{-t}{3}}$ meet at $t=1,3$ .

Explanation of Solution

Given:

The given equation is $y=e^{\frac{-t}{3}}$ .

Calculation:

Consider the required function is,

  $y=e^{\frac{-t}{3}}$

The expression for the above equation is shown in Figure 2

  

Figure 2

The graph of $V$ will touch the graph of $y=e^{\frac{-t}{3}}$ when both the graph will preserve the same value as,

  $V=e^{\frac{-t}{3}}$

Then,

  $\begin{array}{l}{e}^{\frac{-t}{3}}\cos \left(\pi t\right)=e^{\frac{-t}{3}}\\ \cos \left(\pi t\right)=1\\ t=0,2\end{array}$

For, $t=0$

  $\begin{array}{c}V\left(0\right)=y\\ =1\end{array}$

For $t=2$ ,

  $\begin{array}{c}V\left(2\right)=y\\ =e^{\frac{-2}{3}}\\ =0.513\end{array}$

The graph for $y=-e^{\frac{-t}{3}}$ is shown in Figure 3

  

Figure 3

The graph of $V$ will touch the graph of $y=-e^{\frac{-t}{3}}$ when both the graph will preserve the same value as,

  $V=-e^{\frac{-t}{3}}$

Then,

  $\begin{array}{l}{e}^{\frac{-t}{3}}\cos \left(\pi t\right)=-e^{\frac{-t}{3}}\\ \cos \left(\pi t\right)=-1\\ t=1,3\end{array}$

For, $t=0$

  $\begin{array}{c}V\left(1\right)=y\\ =-e^{-\frac{1}{3}}\end{array}$

For $t=3$ ,

  $\begin{array}{c}V\left(3\right)=y\\ =-e^{\frac{-3}{3}}\\ =-0.368\end{array}$

(c)

To determine

To find: Thetime at which the voltage is between $-0.4$ and $4\text{V}$ .

Answer to Problem 51AYU

Thevoltage range of $-0.4<V<0.4$ is obtained at time intervals $0.35<t<0.675$ , $1.29<t<1.75$ and $2.19<t<3$ .

Explanation of Solution

Given:

The voltage is between is between $-0.4$ and $4\text{V}$ .

Calculation:

The graph of the function $V\left(t\right)=e^{\frac{-t}{3}}\cos \left(\pi t\right)$ in the interval between $\left[0,1\right]$ and $\left[-0.4,0.4\right]$ is shown in Figure 4

  

Figure 4

The graph of the function $V\left(t\right)=e^{\frac{-t}{3}}\cos \left(\pi t\right)$ in the interval between $\left[1,2\right]$ and $\left[-0.4,0.4\right]$ is shown in Figure 5

  

Figure 5

The graph of the function $V\left(t\right)=e^{\frac{-t}{3}}\cos \left(\pi t\right)$ in the interval between $\left[1,2\right]$ and $\left[-0.4,0.4\right]$ is shown in Figure 6

  

Figure 6

From the figure 4,5,6 is the voltage range of $-0.4<V<0.4$ is obtained at time intervals $0.35<t<0.675$ , $1.29<t<1.75$ and $2.19<t<3$ .