Question

the diagram shows the aerial view of a park. what is the length of the parks boundary to the nearest yard? use the value π = 3.14.
a. 215 yards
b. 266 yards
c. 285 yards
d. 309 yards

Explanation:

Step1: Calculate the arc length

The park's boundary consists of two arcs. Each arc has a central angle of \(120^{\circ}\) (since \(360 - 60\times2=240^{\circ}\) for the two arcs combined, but wait no - looking at the structure, the boundary is made of two \(120^{\circ}\) arcs. The formula for the arc length \(L=\frac{\theta}{360}\times2\pi r\), where \(\theta\) is the central angle and \(r\) is the radius. Here \(r = 50\) yards.
For one \(120^{\circ}\) arc: \(L_1=\frac{120}{360}\times2\times3.14\times50=\frac{1}{3}\times314=\frac{314}{3}\) yards.
Since there are two \(120^{\circ}\) arcs, the total arc - length \(L_{arc}=2\times\frac{120}{360}\times2\times3.14\times50=\frac{2}{3}\times314\) yards.

Step2: Calculate the length of the straight parts

The straight parts: there are two sides. Each side is \(50\) yards. So the total length of the straight parts \(L_{straight}=2\times50 = 100\) yards.

Step3: Calculate the total boundary length

The total boundary length \(L = L_{arc}+L_{straight}\).
\(L=\frac{2}{3}\times3.14\times100+100\) (since \(2\times50\) in the arc formula and then adding the straight parts).
\(L=\frac{628}{3}+100=\frac{628 + 300}{3}=\frac{928}{3}\approx309.33\) yards.

Answer:

D. 309 yards