Question

determine the shaded area.

the shaded area is
(type an integer or decimal rounded to the nearest hundredth as needed.)

Explanation:

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"concepts_used": [
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<reasoning>

Identify the geometric shapes and dimensions

The shaded region is a semi-annulus, which is the difference between a larger semicircle and a smaller semicircle.

• Outer diameter \(D = 10\text{ in.}\), so outer radius \(R = 5\text{ in.}\)
• Inner diameter \(d = 5\text{ in.}\), so inner radius \(r = 2.5\text{ in.}\)

Calculate the areas of the semicircles

• Outer semicircle area:

\[A_{\text{outer}} = \frac{1}{2} \pi R^2 = \frac{1}{2} \pi (5)^2 = 12.5\pi\]

• Inner semicircle area:

\[A_{\text{inner}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (2.5)^2 = 3.125\pi\]

Compute the shaded area

• Subtract the inner area from the outer area:

\[A_{\text{shaded}} = 12.5\pi - 3.125\pi = 9.375\pi\]

• Convert to a decimal rounded to the nearest hundredth:

\[A_{\text{shaded}} \approx 9.375 \times 3.14159265 \approx 29.4524 \approx 29.45\text{ sq in.}\]
</reasoning>

<answer>
The shaded area is <blank>29.45</blank> <blank>sq in.</blank>
</answer>

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"question_type": "Fill-in-the-blank",
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"Shaded Area"
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Answer:

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<reasoning>

Identify the geometric shapes and dimensions

The shaded region is a semi-annulus, which is the difference between a larger semicircle and a smaller semicircle.

• Outer diameter \(D = 10\text{ in.}\), so outer radius \(R = 5\text{ in.}\)
• Inner diameter \(d = 5\text{ in.}\), so inner radius \(r = 2.5\text{ in.}\)

Calculate the areas of the semicircles

• Outer semicircle area:

\[A_{\text{outer}} = \frac{1}{2} \pi R^2 = \frac{1}{2} \pi (5)^2 = 12.5\pi\]

• Inner semicircle area:

\[A_{\text{inner}} = \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (2.5)^2 = 3.125\pi\]

Compute the shaded area

• Subtract the inner area from the outer area:

\[A_{\text{shaded}} = 12.5\pi - 3.125\pi = 9.375\pi\]

• Convert to a decimal rounded to the nearest hundredth:

\[A_{\text{shaded}} \approx 9.375 \times 3.14159265 \approx 29.4524 \approx 29.45\text{ sq in.}\]
</reasoning>

<answer>
The shaded area is <blank>29.45</blank> <blank>sq in.</blank>
</answer>

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