Question

lesson 8.8 homework
name

1. find the area of the shaded sector. round your answer to the nearest tenth.

(image of a circle with radius 3 cm and central angle 117° for the shaded sector)

1. find the area of the shaded sector. round your answer to the nearest tenth.

(image of a circle with radius 7 in and central angle 141° for the shaded sector)

1. the shaded sector has an area of 24π square inches. find the central angle of the sector, θ.

(image of a circle with radius 12 in and shaded sector with central angle θ)

Explanation:

Problem 1:

Step 1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle is \( A=\frac{\theta}{360}\times\pi r^{2} \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

Step 2: Identify the values of \( \theta \) and \( r \)

From the diagram, \( \theta = 117^{\circ} \) and \( r = 3\space\text{cm} \).

Step 3: Substitute the values into the formula

Substitute \( \theta = 117^{\circ} \) and \( r = 3\space\text{cm} \) into the formula:
\( A=\frac{117}{360}\times\pi\times(3)^{2} \)
First, calculate \( (3)^{2}=9 \). Then, \( \frac{117}{360}\times\pi\times9=\frac{117\times9\pi}{360}=\frac{1053\pi}{360} \)
Simplify \( \frac{1053}{360}=\frac{117}{40} = 2.925 \)
So, \( A = 2.925\pi\approx2.925\times3.1416\approx9.2\space\text{cm}^2 \) (rounded to the nearest tenth)

Step 1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle is \( A=\frac{\theta}{360}\times\pi r^{2} \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle.

Step 2: Identify the values of \( \theta \) and \( r \)

From the diagram, \( \theta = 141^{\circ} \) and \( r = 7\space\text{in} \).

Step 3: Substitute the values into the formula

Substitute \( \theta = 141^{\circ} \) and \( r = 7\space\text{in} \) into the formula:
\( A=\frac{141}{360}\times\pi\times(7)^{2} \)
First, calculate \( (7)^{2}=49 \). Then, \( \frac{141}{360}\times\pi\times49=\frac{141\times49\pi}{360}=\frac{6909\pi}{360} \)
Simplify \( \frac{6909}{360}=\frac{2303}{120}\approx19.1917 \)
So, \( A=\frac{2303}{120}\pi\approx19.1917\times3.1416\approx60.3\space\text{in}^2 \) (rounded to the nearest tenth)

Step 1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle is \( A=\frac{\theta}{360}\times\pi r^{2} \), where \( \theta \) is the central angle in degrees and \( r \) is the radius of the circle. We need to solve for \( \theta \).

Step 2: Identify the values of \( A \) and \( r \)

From the problem, \( A = 24\pi\space\text{square inches} \) and \( r = 12\space\text{in} \).

Step 3: Substitute the values into the formula and solve for \( \theta \)

Substitute \( A = 24\pi \) and \( r = 12\space\text{in} \) into the formula:
\( 24\pi=\frac{\theta}{360}\times\pi\times(12)^{2} \)
First, divide both sides by \( \pi \):
\( 24=\frac{\theta}{360}\times144 \)
Then, solve for \( \theta \):
\( \frac{\theta}{360}=\frac{24}{144}=\frac{1}{6} \)
Multiply both sides by \( 360 \):
\( \theta=\frac{1}{6}\times360 = 60^{\circ} \)

Answer:

The area of the shaded sector is approximately \( 9.2\space\text{cm}^2 \)

Problem 2: