Chapter 13.2, Problem 41PPS

a.

To determine

Draw an angle with the given measure in standard position.

Explanation of Solution

Given:

It is given in the question that a swing has a ${165}^{\circ }$ angle of rotation.

Graph:

  

Interpretation:

In this, use the protractor and place them in the fixed point O and mark an angle that is given in the question in anti clockwise direction i.e. ${165}^{\circ }$ in A and after that draw a line on that marking point A and it is seen in the above that $\alpha ={165}^{\circ }$ .

b.

To determine

Calculate the degree into radian.

Answer to Problem 41PPS

The radian is $\frac{11\pi }{12}$ .

Explanation of Solution

Given:

It is given in the question that the degree is ${165}^{\circ }$ .

Concept Used:

In this, use the concept of multiplying the degree into $\frac{\pi }{180}$ to get the radian.

Calculation: Since, the degree is ${165}^{\circ }$ .

To convert in radian, multiplying by $\frac{\pi }{180}$ ,

  ${165}^{\circ }={165}^{\circ }\times \frac{\pi }{180}=\frac{11\pi }{12}$

Conclusion:

The radian is $\frac{11\pi }{12}$ .

c.

To determine

Find the length of the arc that swings make.

Answer to Problem 41PPS

The arc length of the swing is $18.70ft$ .

Explanation of Solution

Given: It is given in the question that the radius is $6\frac{1}{2}ft$ and the angle is $\frac{11\pi }{12}$ .

Concept Used:

In this, use the concept of arc length formula i.e. $s=r\cdot \theta$

Calculation: Here, $r=6\frac{1}{2}ft,\theta =\frac{11\pi }{12}$

  $\begin{array}{l}s=r\cdot \theta \\ s=6\frac{1}{2}ft\cdot \frac{11\pi }{12}\\ s=\frac{13}{2}ft\cdot \frac{11\pi }{12}\\ s=\frac{143\pi }{24}ft\\ s=\frac{143\times 3.14}{24}ft\\ s=18.70ft\end{array}$

Conclusion:

The arc length is $18.70ft$ .

d.

To determine

Explain that the arc length will change if the lengths of the chains in the swing were doubled.

Answer to Problem 41PPS

The arc length by the swings will doubled.

Explanation of Solution

Given:

It is given in the question that the arc length is $s=r\cdot \theta$ .

Since, the arc length is $s=r\cdot \theta$ .

Here s is directly proportional to the radius r or length of the chains , so if length of the chains increases then it’s arc length will also increases.

So, if the radius will double then it’s arc length will double.

  $s=r\cdot \theta \Rightarrow s=2r\theta \Rightarrow sisdoubled.$