Question

the neighborhood next to a school has a park with a path for walking or biking. because of the shape of the land, the path looks like the diagram below. the radius for both arcs is 50 feet. how far, to the nearest foot, will a walker travel in one lap on this path? 1 of 4 questio 683 ft 558 ft 185 ft 592 ft

Explanation:

Step1: Calculate the length of the \(120^{\circ}\) arc

The formula for the length of an arc is \(L=\frac{\theta}{360}\times2\pi r\). For \(\theta = 120^{\circ}\) and \(r = 50\) ft, \(L_1=\frac{120}{360}\times2\pi\times50=\frac{1}{3}\times100\pi=\frac{100\pi}{3}\) ft.

Step2: Calculate the length of the \(90^{\circ}\) arc

Using the same formula, for \(\theta = 90^{\circ}\) and \(r = 50\) ft, \(L_2=\frac{90}{360}\times2\pi\times50=\frac{1}{4}\times100\pi = 25\pi\) ft.

Step3: Calculate the total length of the path

The total length \(L\) of the path is the sum of the two arc - lengths and the two straight - line segments (each of length \(250\) ft). So \(L=\frac{100\pi}{3}+25\pi + 250+250\).
First, find a common denominator for the arc - length terms: \(\frac{100\pi}{3}+25\pi=\frac{100\pi}{3}+\frac{75\pi}{3}=\frac{100\pi + 75\pi}{3}=\frac{175\pi}{3}\).
Then, \(L=\frac{175\pi}{3}+500\).
Substitute \(\pi\approx3.14\): \(L=\frac{175\times3.14}{3}+500\).
\(175\times3.14 = 549.5\), so \(\frac{549.5}{3}+500\approx183.17+500=683.17\approx683\) ft.

Answer:

683 ft