Question

what is the area of the composite figure?
○ ((6pi + 10), \text{m}^2)
○ ((10pi + 10), \text{m}^2)
○ ((12pi + 10), \text{m}^2)
○ ((16pi + 10), \text{m}^2)

Explanation:

Answer:

First, analyze the composite figure: it consists of a semicircular ring (annulus) and a rectangle.

1. Semicircular Ring (Annulus):

• Inner diameter = 4 m ⇒ Inner radius \( r = \frac{4}{2} = 2 \) m.
• Outer radius \( R = 2 + 2 = 4 \) m (since the width of the ring is 2 m).
• Area of a full annulus: \( \pi(R^2 - r^2) = \pi(4^2 - 2^2) = \pi(16 - 4) = 12\pi \).
• Area of the semicircular ring: \( \frac{1}{2} \times 12\pi = 6\pi \).

1. Rectangle:

• Dimensions: 2 m (width) × 5 m (height) ⇒ Area = \( 2 \times 5 = 10 \) m².

1. Total Area:

Sum of the semicircular ring and rectangle: \( 6\pi + 10 \) m².

So the correct option is: \( \boldsymbol{(6\pi + 10)\ \text{m}^2} \) (the first option).