Question

1. r = 5 in.

Explanation:

1. Find the area of the circle:

• The formula for the area of a circle is \(A_{circle}=\pi r^{2}\), where \(r = 5\) inches.
• Substitute \(r = 5\) into the formula: \(A_{circle}=\pi\times5^{2}=25\pi\) square - inches.

1. Find the area of the triangle:

• The formula for the area of a triangle is \(A_{triangle}=\frac{1}{2}\times base\times height\). Here, the base is \(2\times3 = 6\) inches (since the perpendicular from the center to the chord bisects the chord) and the height is 4 inches.
• Substitute the values into the formula: \(A_{triangle}=\frac{1}{2}\times6\times4=12\) square - inches.

1. Find the area of the shaded region:

• The area of the shaded region \(A\) is the area of the circle minus the area of the triangle.
• \(A = 25\pi-12\).
• Using an approximation of \(\pi\approx3.14\), we have \(A\approx25\times3.14 - 12=78.5 - 12 = 66.5\) square - inches.

Step1: Calculate circle area

Use formula \(A_{circle}=\pi r^{2}\), with \(r = 5\). So \(A_{circle}=25\pi\).

Step2: Calculate triangle area

Base is \(6\) (from \(2\times3\)), height is \(4\). Use \(A_{triangle}=\frac{1}{2}\times base\times height\), so \(A_{triangle}=12\).

Step3: Calculate shaded - region area

Subtract triangle area from circle area. \(A = 25\pi-12\approx66.5\).

Answer:

\(25\pi - 12\approx66.5\) square inches.