Question

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

Explanation:

Step1: Analyze the composite figure

The composite figure consists of a semicircular ring (annulus) and a rectangle. First, find the radii of the two semicircles. The inner semicircle has a diameter of 4 m, so its radius \( r = \frac{4}{2}=2 \) m. The outer semicircle has a radius \( R = 2 + 2 = 4 \) m (since the width of the ring is 2 m).

Step2: Area of the semicircular ring

The area of a full annulus is \( \pi(R^2 - r^2) \), so the area of the semicircular ring is \( \frac{1}{2}\pi(R^2 - r^2) \). Substitute \( R = 4 \) and \( r = 2 \):
\[ \frac{1}{2}\pi(4^2 - 2^2)=\frac{1}{2}\pi(16 - 4)=\frac{1}{2}\pi(12) = 6\pi \] Wait, no, wait. Wait, maybe I made a mistake. Wait, the outer radius: the inner diameter is 4, so inner radius is 2. The width is 2, so outer radius is 2 + 2 = 4? Wait, no, the horizontal segment is 4 m (diameter of inner semicircle), and the width of the ring is 2 m, so the outer diameter is 4 + 2*2 = 8 m? Wait, no, the figure: the inner semicircle has a diameter of 4 m (so radius 2 m), and the ring has a width of 2 m, so the outer semicircle has a radius of 2 + 2 = 4 m? Wait, no, maybe the outer radius is (4/2)+2 = 2 + 2 = 4? Wait, no, let's re-examine the figure. The inner semicircle has a diameter of 4 m (so length 4 m), and the horizontal part of the rectangle is 2 m? Wait, the rectangle has a height of 5 m and width of 2 m? Wait, no, the rectangle: the vertical side is 5 m, and the horizontal side is 2 m? Wait, the composite figure: the lower part is a rectangle with length 5 m and width 2 m? Wait, no, the rectangle: the width is 2 m (the same as the ring width) and height 5 m? Wait, maybe I misread. Let's re-express:

Wait, the composite figure: the curved part is a semicircular ring (outer semicircle minus inner semicircle), and the straight part is a rectangle. Let's find the area of the semicircular ring:

Inner radius \( r = \frac{4}{2}=2 \) m.

Outer radius \( R = 2 + 2 = 4 \) m (since the ring width is 2 m).

Area of semicircular ring: \( \frac{1}{2}\pi(R^2 - r^2)=\frac{1}{2}\pi(16 - 4)=\frac{1}{2}\pi(12)=6\pi \)? Wait, no, that can't be. Wait, maybe the outer radius is (4/2)+2 = 2 + 2 = 4, but the outer diameter would be 8, so outer semicircle area is \( \frac{1}{2}\pi(4)^2 = 8\pi \), inner semicircle area is \( \frac{1}{2}\pi(2)^2 = 2\pi \), so the ring area is \( 8\pi - 2\pi = 6\pi \)? Wait, no, \( \frac{1}{2}\pi R^2 - \frac{1}{2}\pi r^2 = \frac{1}{2}\pi(R^2 - r^2) \). So \( R = 4 \), \( r = 2 \), so \( \frac{1}{2}\pi(16 - 4) = 6\pi \). Then the rectangle: the rectangle has a length of 5 m and width of 2 m? Wait, no, the rectangle: the vertical side is 5 m, and the horizontal side is 2 m? Wait, the rectangle's area is length * width = 5 * 2 = 10 m². Wait, but then the semicircular ring area: wait, maybe I made a mistake in the radii. Wait, the inner semicircle has a diameter of 4 m (so radius 2 m), and the outer semicircle has a diameter of 4 + 2*2 = 8 m (so radius 4 m). Then the area of the outer semicircle is \( \frac{1}{2}\pi(4)^2 = 8\pi \), inner semicircle is \( \frac{1}{2}\pi(2)^2 = 2\pi \), so the ring area is \( 8\pi - 2\pi = 6\pi \). Then the rectangle: the rectangle has dimensions 2 m (width) and 5 m (height), so area 2*5 = 10 m². Wait, but that would be 6π + 10, but that's one of the options. Wait, but the options include (10π + 10), (12π + 10), etc. Wait, maybe my outer radius is wrong. Wait, maybe the outer radius is (4/2) + 2 = 2 + 2 = 4, but the outer diameter is 8, so outer semicircle area is \( \frac{1}{2}\pi(4)^2 = 8\pi \), inner is \( \frac{1}{2}\pi(2)^2 = 2\pi \), difference is 6π. Then the rectangle: 2*5=10. So total area 6π + 10? But that's the first option. Wait, but maybe I messed up the ring. Wait, maybe the outer radius is (4 + 2*2)/2 = (8)/2 = 4, inner radius 2. Wait, no, let's check the options. The options are (6π + 10), (10π + 10), (12π + 10), (16π + 10). Wait, maybe my ring area is wrong. Wait, maybe the outer radius is 4 m? Wait, no, maybe the inner diameter is 4, so inner radius 2, and the outer radius is 2 + 2 = 4? No, that would make the outer diameter 8, but maybe the outer radius is 4, and the inner radius is 2, but the area of the semicircular ring is \( \frac{1}{2}\pi(4^2 - 2^2) = 6\pi \), and the rectangle area is 2*5=10, so total 6π + 10, which is the first option. But wait, maybe the rectangle is 2*5=10, and the ring is 6π, so total 6π + 10. So the answer should be (6π + 10) m².

Wait, but let's recheck. The inner semicircle: diameter 4 m, so radius 2 m. Outer semicircle: radius 2 + 2 = 4 m (since the ring width is 2 m). Area of outer semicircle: \( \frac{1}{2}\pi(4)^2 = 8\pi \). Area of inner semicircle: \( \frac{1}{2}\pi(2)^2 = 2\pi \). Area of semicircular ring: 8π - 2π = 6π. Area of rectangle: length 5 m, width 2 m (since the ring width is 2 m), so 2*5 = 10 m². Total area: 6π + 10 m². So the first option.

Step3: Area of the rectangle

The rectangle has a width of 2 m (same as the ring width) and a height of 5 m. So area \( A_{rectangle} = 2 \times 5 = 10 \) m².

Step4: Total area of composite figure

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

Answer:

\( (6\pi + 10) \, \text{m}^2 \) (the first option)