Question

what is the area of the shaded portion of the circle?
(25π − 5.8) ft²
(5π − 11.6) ft²
(25π − 11.6) ft²
(5π − 5.8) ft²

Explanation:

Step1: Find the area of the sector

The formula for the area of a sector of a circle is \( A_{sector} = \frac{\theta}{360^\circ} \times \pi r^2 \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle. Here, \( \theta = 72^\circ \) and \( r = 5 \) ft.

$$ A_{sector} = \frac{72^\circ}{360^\circ} \times \pi \times (5)^2 = \frac{1}{5} \times 25\pi = 5\pi $$

Step2: 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 \( 2.9 + 2.9 = 5.8 \) ft (since the two equal sides are 2.9 ft each) and the height is 4 ft.

$$ A_{triangle} = \frac{1}{2} \times 5.8 \times 4 = 5.8 \times 2 = 11.6 $$

Step3: Find the area of the shaded region

The area of the shaded region is the area of the sector minus the area of the triangle.

$$ A_{shaded} = A_{sector} - A_{triangle} = 5\pi - 11.6 $$

Answer:

\( (5\pi - 11.6) \text{ft}^2 \) (the second option: \( (5\pi - 11.6) \text{ft}^2 \))