Question

iii. using arc length or sector area
9.) find the perimeter of the shaded region.

Explanation:

Step1: Find the length of the arc

The formula for the length of an arc \(l=\frac{\theta}{360}\times2\pi r\). Here, \(\theta = 90^{\circ}\) (since the central angle of the sector is \(90^{\circ}\)) and \(r = 5\) ft.

$$l=\frac{90}{360}\times2\pi\times5=\frac{1}{4}\times10\pi=\frac{5\pi}{2}\text{ ft}$$

Step2: Find the length of the two radii segments

The two radii segments are of equal length. Each radius \(r = 5\) ft, and the sum of their lengths \(s=5 + 5=10\) ft

Step3: Calculate the perimeter \(P\) of the shaded region

The perimeter \(P\) of the shaded region is the sum of the arc length and the two radii segments.

$$P=\frac{5\pi}{2}+10$$

Substitute \(\pi\approx3.14\)

$$P=\frac{5\times3.14}{2}+10=7.85 + 10=17.85\text{ ft}$$

Answer:

\(17.85\) ft