Question

The semicircle of area 50 centimeters is inscribed inside a rectangle. The diameter of the semicircle coincides with the length of the rectangle. Find the area of the rectangle.

A) 25 square centimeters
B) 50 square centimeters
C) 100 square centimeters
D) 200 square centimeters

Answer 1

The area of the rectangle is 100 square centimeters (option C).

Step-by-step explanation:

The area of a smicircle is half the area of a full circle with the same diameter. Given that the area of the semicircle is 50 square centimeters, we can determine the area of the full circle using the formula for the area of a circle, which is
`\(A_{\text{circle}} = \pi r^2\),` where ris the radius.

Since the semicircle's diameter coincides with the length of the rectangle, the diameter of the semicircle equals the length of the rectangle. Thus, the radius of the semicircle is half the length of the rectangle.

From the formula for the area of a circle, if the area of the semicircle is 50 square centimeters, then the area of the full circle is
`\(2 * 50 = 100\)`square centimeters.

The area of the rectangle can be determined as the length multiplied by the width. Given that the length of the rectangle is the diameter of the semicircle, which corresponds to the diameter of the full circle, and the width is the radius of the semicircle, the area of the rectangle is equal to the area of the full circle, which is 100 square centimeters. Therefore, the correct option representing the area of the rectangle is \(100\) square centimeters (option C).