Question

a square lawn has area ( 128 mathrm{ft}^{2} ). a sprinkler placed at the center of the lawn sprays water in a circular pattern as shown in the figure. what is the radius of the circle? the radius of the circle is ( square mathrm{ft} ). (simplify your answer. use a comma to separate answers as needed.)

Explanation:

Step1: Find the side length of the square

The area formula of a square is \(A = s^{2}\), where \(A\) is the area and \(s\) is the side - length. Given \(A=128\space ft^{2}\), then \(s=\sqrt{128}=\sqrt{64\times2}=8\sqrt{2}\space ft\).

Step2: Relate the side - length of the square to the radius of the circle

The diagonal of the square is equal to the diameter \(d\) of the circle. The diagonal of a square with side - length \(s\) is \(d = s\sqrt{2}\). Substituting \(s = 8\sqrt{2}\), we get \(d=8\sqrt{2}\times\sqrt{2}=16\space ft\).
Since the radius \(r=\frac{d}{2}\), then \(r = 8\space ft\).

Answer:

\(8\)