Question

Given: BC = 10 inches AC = inches m∠CBD = 60° m∠CAD = 90° Calculate the exact area of the shaded region. Show all work for full credit.

Answer 1

Answer:

23.3285 in²

Step-by-step explanation:

From the above diagram, we can see that the shaded portion is a circular in shape. The shaded portion is coming together of two segments of two circles

The formula for the area of the segment is given as:

A = (½) × r² × [(π/180) θ – sin θ]

 Where,

 r: radius of the circle

θ = angle

a) Circle 1: the largest circle:

A = (½) × r² × [(π/180) θ – sin θ]

 r: radius of the circle = 10 inches

θ = angle = 60°

A = (½) × 10² × [(π/180) × 60° – sin 60]

A = 9.0586 in²

b) Circle 2: the smallest circle:

A = (½) × r² × [(π/180) θ – sin θ]

 r: radius of the circle = √50 inches

θ = angle = 90°

A = (½) × (√50) ² × [(π/180) × 90° – sin 90]

A = 14.2699 in²

 Total area of the shaded portion:

Area of the segment of the large circle + Area of the segment of the small circle

A = 9.0586 in² + 14.2699 in²

A = 23.3285 in²