Chapter 9.2, Problem 9GP

Solution

To find: the arc length of the outfield fence and the area of the field.

Length: $s=345.57\text{ feet}$

Area: $A=38013.27{\text{ feet}}^2$

Given information:

Given softball field forms a sector having radius 220 feet and the central angle $\theta =90°$ .

Definition Used:

Arc length and the area of a sector:

  

The arc length s and area A of a sector with radius r and central angle $\theta$ (measured in radians) are as follows.

Arc length: $s=r\theta$

Area: $A=\frac{1}{2}{r}^2\theta$

Property Used:

Degree can be converted into radian using:

  $1°=\frac{\pi }{180°}radian$

Explanation:

Here, the measure of central angle is given in degrees but as per the above definition this angle is required in radians. So, first step is to convert $\theta =90°$ to radians.

To convert $\theta =115°$ into radians, multiply it by $\frac{\pi }{180°}$ .

  $\begin{array}{c}90°=90°\times \frac{\pi }{180°}\\ =\frac{\pi }{2}\text{ radian}\end{array}$

Now, use the formulas given in above definition to find the length of fence and the area of the field.

Here, use radius $r=220\text{}\text{ feet}$ and central angle $\theta =\frac{\pi }{2}\text{ radian}$

Length of outfield fence:

  $\begin{array}{c}s=r\theta \\ =220\times \frac{\pi }{2}\\ =345.57\text{ feet}\end{array}$

So, the length of outfield fence is $s=345.57\text{ feet}$ .

Area:

  $\begin{array}{c}A=\frac{1}{2}{r}^2\theta \\ =\frac{1}{2}\times {220}^2\times \frac{\pi }{2}\\ =38013.27{\text{ feet}}^2\end{array}$

So, the area of field is $A=38013.27{\text{ feet}}^2$ .