Question

determine the shaded area.

the shaded area is
(type an integer or decimal roun

Explanation:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Area of Circle",
"Shaded Area"
],
"new_concepts": [],
"current_concepts": [
"Area of Circle",
"Shaded Area",
"Semicircle Area"
]
}
</pre_analysis>

<reasoning>

Identify the geometric shapes and dimensions

The shaded region is a half-ring (semicircular annulus).

• 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 outer and inner semicircles

\[

$$\begin{aligned} A_{\text{outer}} &= \frac{1}{2} \pi R^2 = \frac{1}{2} \pi (5)^2 = 12.5\pi \\ A_{\text{inner}} &= \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (2.5)^2 = 3.125\pi \end{aligned}$$

\]

Compute the shaded area

\[

$$\begin{aligned} A_{\text{shaded}} &= A_{\text{outer}} - A_{\text{inner}} \\ &= 12.5\pi - 3.125\pi = 9.375\pi \\ &\approx 9.375 \times 3.14159265 \approx 29.45 \end{aligned}$$

\]
</reasoning>

<answer>
Determine the shaded area.

The shaded area is <blank>\(9.375\pi\)</blank> (or approximately <blank>\(29.45\)</blank>) \(\text{in.}^2\).
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area of Semicircle and Annulus"
]
}
</post_analysis>

Answer:

<pre_analysis>
{
"quality": "clear",
"question_count": 1,
"skills_matched": [
"step_cot",
"blank_filling_formatter",
"explain_then_answer",
"quick"
],
"needs_drawing": false,
"concepts_used": [
"Area of Circle",
"Shaded Area"
],
"new_concepts": [],
"current_concepts": [
"Area of Circle",
"Shaded Area",
"Semicircle Area"
]
}
</pre_analysis>

<reasoning>

Identify the geometric shapes and dimensions

The shaded region is a half-ring (semicircular annulus).

• 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 outer and inner semicircles

\[

$$\begin{aligned} A_{\text{outer}} &= \frac{1}{2} \pi R^2 = \frac{1}{2} \pi (5)^2 = 12.5\pi \\ A_{\text{inner}} &= \frac{1}{2} \pi r^2 = \frac{1}{2} \pi (2.5)^2 = 3.125\pi \end{aligned}$$

\]

Compute the shaded area

\[

$$\begin{aligned} A_{\text{shaded}} &= A_{\text{outer}} - A_{\text{inner}} \\ &= 12.5\pi - 3.125\pi = 9.375\pi \\ &\approx 9.375 \times 3.14159265 \approx 29.45 \end{aligned}$$

\]
</reasoning>

<answer>
Determine the shaded area.

The shaded area is <blank>\(9.375\pi\)</blank> (or approximately <blank>\(29.45\)</blank>) \(\text{in.}^2\).
</answer>

<post_analysis>
{
"subject": "Mathematics",
"question_type": "Fill-in-the-blank",
"knowledge_point": [
"Mathematics",
"Geometry",
"Area of Semicircle and Annulus"
]
}
</post_analysis>