determine the shaded area.

the shaded area is
(type an integer or decimal roun

Question

determine the shaded area.

the shaded area is
(type an integer or decimal roun

Accepted Answer

{ "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" ] } ## 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 $$ \begin{aligned} A_{\text{shaded}} &= A_{\text{outer}} - A_{\text{inner}} \ &= 12.5\pi - 3.125\pi = 9.375\pi \ &\approx 9.375 \times 3.14159265 \approx 29.45 \end{aligned} $$ Determine the shaded area. The shaded area is $9.375\pi$ (or approximately $29.45$) $\text{in.}^2$. { "subject": "Mathematics", "question_type": "Fill-in-the-blank", "knowledge_point": [ "Mathematics", "Geometry", "Area of Semicircle and Annulus" ] }

Sovi.AI - AI Math Tutor

QUESTION IMAGE

Question

Explanation:

Identify the geometric shapes and dimensions

Calculate the areas of the outer and inner semicircles

Compute the shaded area

Answer: