Chapter 7, Problem 7CRT

(a)

To determine

To represent: The function $h$ in the form of $h\left(t\right)=a+b\cos kt$ .

Answer to Problem 7CRT

The function is $h\left(t\right)=45-40\cos \left(8\pi t\right)$ .

Explanation of Solution

Given information:

The radius of the wheel is 40 centimeters and the center of the wheel is 45 centimeters above the floor. The rotation of the wheel is 4 rotations per minute.

Calculation:

Consider the given equation of the function of height.

  $h\left(t\right)=a+b\cos kt$  ... (1)

Where,

• $a$ is the vertical shift.
• $\left|b\right|$ is the amplitude.

When $t=0$ , the point $P$ should be lowest point to travel and it represents a minimum of the function when $b<0$ . Therefore, the value of the radius of the wheel is $-40$ .

Since the rotation of the wheel is 4 rotations per minute. Calculate the value of the constant.

  $\begin{array}{c}\frac{2\pi }{k}=\frac{1}{4}\\ k=8\pi \end{array}$

Substitute the values in equation (1).

  $\begin{array}{c}h\left(t\right)=45+\left(-40\right)\cos 8\pi t\\ h\left(t\right)=45-40\cos 8\pi t\end{array}$

Conclusion:

Therefore, the function is $h\left(t\right)=45-40\cos \left(8\pi t\right)$ .

(b)

To determine

To find: The distance between $B$ and $C$ .

Answer to Problem 7CRT

The distance between $B$ and $C$ is

  $\text{43}\text{.6}\text{cm}$ .

Explanation of Solution

Given information:

The length of the support struts $AB$ and $AC$ is 50 centimeters. Consider the given figure.

  

Calculation:

Since the point $A$ is 45 centimeters above the floor. Let the distance between $B$ and $C$ is $2x$ . Apply the Pythagoras theorem to calculate the value of $x$ .

  $\begin{array}{c}{x}^2={50}^2-{45}^2\\ x=\sqrt{2500-2025}\\ x=\sqrt{475}\text{cm}\end{array}$

Calculate the distance between $B$ and $C$ .

  $\begin{array}{c}2x=2\sqrt{475}\text{cm}\\ \approx \text{43}\text{.6}\text{cm}\end{array}$

Conclusion:

Therefore, the distance between $B$ and $C$ is $\text{43}\text{.6}\text{cm}$ .