Calculate the area of the shaded portion of an annulus with inner radius 4 cm and outer radius 7 cm, where…

Question

Calculate the area of the shaded portion of an annulus with inner radius 4 cm and outer radius 7 cm, where the annulus is divided into 8 equal sectors and 4 are shaded.

ScanSolve AI Tutor · Accepted Answer

Here's how to find the area of the shaded portion:

The figure shows an annulus (a ring shape) formed by two concentric circles. The annulus is divided into 8 equal sectors, with 4 of them shaded.

Identify the radii of the circles.
The inner radius is $r_1 = 4 \text{ cm}$ .
The outer radius is $r_2 = 7 \text{ cm}$ .
Calculate the area of the annulus.
The area of the annulus is the area of the outer circle minus the area of the inner circle.
The formula for the area of a circle is $A = \pi r^2$ .
$A_{annulus} = \pi r_2^2 - \pi r_1^2$ $A_{annulus} = \pi (r_2^2 - r_1^2)$
Substitute the given radii:
$A_{annulus} = \pi ((7 \text{ cm})^2 - (4 \text{ cm})^2)$ $A_{annulus} = \pi (49 \text{ cm}^2 - 16 \text{ cm}^2)$ $A_{annulus} = 33\pi \text{ cm}^2$
Determine the fraction of the annulus that is shaded.
The annulus is divided into 8 equal sectors, and 4 of these sectors are shaded.
The shaded fraction is $\frac{4}{8} = \frac{1}{2}$ .
Calculate the area of the shaded portion.
Multiply the area of the annulus by the shaded fraction.
$A_{shaded} = \frac{1}{2} \times A_{annulus}$ $A_{shaded} = \frac{1}{2} \times 33\pi \text{ cm}^2$ $A_{shaded} = \frac{33}{2}\pi \text{ cm}^2$
Using the approximation $\pi = \frac{22}{7}$ :
$A_{shaded} = \frac{33}{2} \times \frac{22}{7} \text{ cm}^2$ $A_{shaded} = 33 \times \frac{11}{7} \text{ cm}^2$ $A_{shaded} = \frac{363}{7} \text{ cm}^2$
The area of the shaded portion is $\frac{363}{7} \text{ cm}^2$ .

Calculate the area of the shaded portion of an annulus with inner radius 4 cm and outer radius 7 cm, where the annulus is divided into 8 equal sectors and 4 are shaded.

Answer

Step-by-step solution

Identify the radii of the circles.

Calculate the area of the annulus.

Determine the fraction of the annulus that is shaded.

Calculate the area of the shaded portion.

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