Chapter 4.2, Problem 77E

(a)

To determine

Find the displace $\text{y(t)}=\frac{1}{4}\cos \text{6t}$ when t = 0

Answer to Problem 77E

  $0.25$

Explanation of Solution

Given: The function: $\text{y(t)}=\frac{1}{4}\cos \text{6t}$

Where y is the displacement (in feet) and t is the time in sec.

Concept Used:

Cosine function is an Even function and periodic with a period $2\text{π}$

Calculation:When t = 0

  $\begin{array}{l}\text{t}=0\\ \text{y(t)}=\frac{1}{4}\cos \text{6t}=\frac{1}{4}\cos \text{6(0)}\text{=}\frac{1}{4}\cos \text{0}\text{=}\frac{1}{4}[\cos \text{0}\text{=}\text{1}\text{] }\\ \text{y(t)}=\frac{1}{4}=0.25\end{array}$

Thus, when $\text{t}=0$ , the displacement $\text{y(t)}=0.25$

(b)

To determine

Find the displacement $\text{y(t)}=\frac{1}{4}\cos \text{6t}$when $\text{t}\text{=}\frac{1}{4}$

Answer to Problem 77E

  $-0.2278$

Explanation of Solution

Given:

The function: $\text{y(t)}=\frac{1}{4}\cos \text{6t}$

Where y is the displacement (in feet) and t is the time in sec.

Concept Used:

Cosine function is an Even function and periodic with a period $2\text{π}$

Calculation: When $\text{t}=\frac{1}{4}$

  $\begin{array}{l}\text{t}=\frac{1}{4}\\ \text{y(t)}=\frac{1}{4}\cos \text{6t}=\frac{1}{4}\cos \text{6(}\frac{1}{4}\text{)}\text{=}\frac{1}{4}\cos \frac{3}{2}\text{=}-0.2278\\ \text{y(t)}=-0.2278\end{array}$

Thus, when $\text{t}=\frac{1}{4}$ , the displacement $\text{y(t)}=-0.2278$

(c)

To determine

Find the displacement $\text{y(t)}=\frac{1}{4}\cos \text{6t}$ when $\text{t}\text{=}\frac{1}{2}$

Answer to Problem 77E

  $0.1651$

Explanation of Solution

Given: The function: $\text{y(t)}=\frac{1}{4}\cos \text{6t}$

Where y is the displacement (in feet) and t is the time in sec.

Concept Used:

Cosine function is an Even function and periodic with a period $2\text{π}$

Calculation:When $\text{t}=\frac{1}{2}$

  $\begin{array}{l}\text{t}=\frac{1}{2}\\ \text{y(t)}=\frac{1}{4}\cos \text{6t}=\frac{1}{4}\cos \text{6(}\frac{1}{2}\text{)}\text{=}\frac{1}{4}\cos 3\text{=}0.1651\\ \text{y(t)}=0.1651\end{array}$

Thus, when $\text{t}=\frac{1}{2}$ , the displacement $\text{y(t)}=0.1651$