Question

1. the figure below shows four congruent semicircles that touch only at the corners of a square. the path from point x, along the diameters of the semicircles, back to point x is 96 inches long. how long is the path from point x, along the arcs of the semicircles, back to point x, in inches? f. 24π g. 48π h. 96π j. 144π k. 288π

Explanation:

Step1: Find the diameter of each semicircle

Let the diameter of each semicircle be \(d\). The path along the diameters consists of \(4\) diameters. So \(4d = 96\), then \(d=\frac{96}{4}=24\) inches.

Step2: Calculate the length of the path along the arcs

The path along the arcs is composed of \(4\) semicircles. The length of a semicircle is \(l=\frac{1}{2}\times\pi\times d\). Since there are \(4\) semicircles, the total length \(L = 4\times\frac{1}{2}\times\pi\times d\). Substitute \(d = 24\) into the formula, we get \(L=2\times\pi\times24 = 48\pi\) inches.

Answer:

G. \(48\pi\)