Chapter 9.3, Problem 1MRPS

Solution

a.

To state: The distance from the goal line to the person where the person will leave the circle if the band members form a circle 100 feet wide in the center of the field. A person starts at a point on the circle 100 feet from the goal line, march ${300}^{\circ }$ around the circle, and then walk towards the goal line exit the field.

  

The resultant answer is 125 feet.

Given information:

The band members form a circle 100 feet wide in the center of the field. A person starts at a point on the circle 100 feet from the goal line, march ${300}^{\circ }$ around the circle, and then walk towards the goal line exit the field.

Explanation:

First draw the diagram using the given information:

  

From the above figure, the distance where the person leaves the circle can be find out:

  $100+\left(50-x\right)$

Now find x , using the right triangle having 50 as the hypotenuse and x as one of the sides. Write a trigonometric equation involving x .

  $\cos {60}^{\circ }=\frac{x}{50}$

Now solve for x ,

  $\begin{array}{l}x=50\cos {60}^{\circ }\\ x=25\end{array}$

Now evaluate $100+\left(50-x\right)$ ,

  $\begin{array}{c}100+\left(50-x\right)=100+\left(50-25\right)\\ =100+25\\ =125\end{array}$

Therefore, the point where the person leaves the circle is 125 feet far from the goal line.

b.

To state: The distance for which the person march around the circle.

The resultant answer is 262 feet.

Given information:

The band members form a circle 100 feet wide in the center of the field. A person starts at a point on the circle 100 feet from the goal line, march ${300}^{\circ }$ around the circle, and then walk towards the goal line exit the field.

Explanation:

The diagram is:

  

It can be seen that the distance walked by the person is equivalent to the length of the arc that subtends the central angle ${300}^{\circ }$ .

The length of the arc is given by $s=r\theta$ , where r is the radius and $\theta$ is the angle subtended in radians by the arc at the center.

Now replace $r=50\text{ and }\theta ={300}^{\circ }\left(\frac{\pi }{{180}^{\circ }}\text{ radians}\right)$ and evaluate it,

  $\begin{array}{c}s=50\cdot {300}^{\circ }\left(\frac{\pi }{{180}^{\circ }}\text{ radians}\right)\\ =50\cdot \frac{15\pi }{9}\\ \approx 262\end{array}$

Therefore, the distance marched by the person is about 262 feet around the circle.