QuestionFind the length of arc UV in a circle sector with area $81 \pi$ . Choices: (A) $6 \pi$ , (B) $9 \pi$ , (C) $27 \pi$ , (D) $54 \pi$ , (E) $108 \pi$ .

Studdy Solution

STEP 1

Assumptions1. The sector of the circle has an area of $81 \pi$ . . We need to find the length of arc UV.

STEP 2

The area of a sector of a circle is given by the formula $Area = \frac{r^2 \times \theta}{2}$ where $r$ is the radius of the circle and $\theta$ is the angle of the sector in radians.

STEP 3

We know that the area of the sector is $81 \pi$ . We can equate this to the formula for the area of a sector and solve for $r$ .
$81 \pi = \frac{r^2 \times \theta}{2}$

STEP 4

We can simplify the equation by multiplying both sides by2.
$162 \pi = r^2 \times \theta$

STEP 5

The angle $\theta$ for a full circle in radians is $2\pi$ . Since the sector is a quarter of a circle, $\theta$ is $\frac{2\pi}{4} = \frac{\pi}{2}$ .

STEP 6

Substitute $\theta$ into the equation and solve for $r$ .
$162 \pi = r^2 \times \frac{\pi}{2}$

STEP 7

implify the equation by multiplying both sides by $\frac{2}{\pi}$ .
$324 = r^2$

STEP 8

Take the square root of both sides to solve for $r$ .
$r = \sqrt{324} =18$

STEP 9

The length of an arc is given by the formula $Arc\, length = r \times \theta$

STEP 10

Substitute the values of $r$ and $\theta$ into the formula to find the length of arc UV.
$Arc\, length =18 \times \frac{\pi}{2}$

STEP 11

Calculate the length of arc UV.
$Arc\, length =18 \times \frac{\pi}{} =9 \pi$ The length of arc UV is $9 \pi$ .

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QuestionFind the length of arc UV in a circle sector with area $81 \pi$ . Choices: (A) $6 \pi$ , (B) $9 \pi$ , (C) $27 \pi$ , (D) $54 \pi$ , (E) $108 \pi$ .

Studdy Solution

STEP 1

Assumptions1. The sector of the circle has an area of $81 \pi$ . . We need to find the length of arc UV.

STEP 2

The area of a sector of a circle is given by the formula $Area = \frac{r^2 \times \theta}{2}$ where $r$ is the radius of the circle and $\theta$ is the angle of the sector in radians.

STEP 3

We know that the area of the sector is $81 \pi$ . We can equate this to the formula for the area of a sector and solve for $r$ .
$81 \pi = \frac{r^2 \times \theta}{2}$

STEP 4

We can simplify the equation by multiplying both sides by2.
$162 \pi = r^2 \times \theta$

STEP 5

The angle $\theta$ for a full circle in radians is $2\pi$ . Since the sector is a quarter of a circle, $\theta$ is $\frac{2\pi}{4} = \frac{\pi}{2}$ .

STEP 6

Substitute $\theta$ into the equation and solve for $r$ .
$162 \pi = r^2 \times \frac{\pi}{2}$

STEP 7

implify the equation by multiplying both sides by $\frac{2}{\pi}$ .
$324 = r^2$

STEP 8

Take the square root of both sides to solve for $r$ .
$r = \sqrt{324} =18$

STEP 9

The length of an arc is given by the formula $Arc\, length = r \times \theta$

STEP 10

Substitute the values of $r$ and $\theta$ into the formula to find the length of arc UV.
$Arc\, length =18 \times \frac{\pi}{2}$

STEP 11

Calculate the length of arc UV.
$Arc\, length =18 \times \frac{\pi}{} =9 \pi$ The length of arc UV is $9 \pi$ .

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QuestionFind the length of arc UV in a circle sector with area 81π81 \pi81π. Choices: (A) 6π6 \pi6π, (B) 9π9 \pi9π, (C) 27π27 \pi27π, (D) 54π54 \pi54π, (E) 108π108 \pi108π.

Studdy Solution

STEP 1

Assumptions1. The sector of the circle has an area of 81π81 \pi81π. . We need to find the length of arc UV.

STEP 2

The area of a sector of a circle is given by the formulaArea=r2×θ2Area = \frac{r^2 \times \theta}{2}Area=2r2×θ​where rrr is the radius of the circle and θ\thetaθ is the angle of the sector in radians.

STEP 3

We know that the area of the sector is 81π81 \pi81π. We can equate this to the formula for the area of a sector and solve for rrr.81π=r2×θ281 \pi = \frac{r^2 \times \theta}{2}81π=2r2×θ​

STEP 4

We can simplify the equation by multiplying both sides by2.162π=r2×θ162 \pi = r^2 \times \theta162π=r2×θ

STEP 5

The angle θ\thetaθ for a full circle in radians is 2π2\pi2π. Since the sector is a quarter of a circle, θ\thetaθ is 2π4=π2\frac{2\pi}{4} = \frac{\pi}{2}42π​=2π​.

STEP 6

Substitute θ\thetaθ into the equation and solve for rrr.162π=r2×π2162 \pi = r^2 \times \frac{\pi}{2}162π=r2×2π​

STEP 7

implify the equation by multiplying both sides by 2π\frac{2}{\pi}π2​.324=r2324 = r^2324=r2

STEP 8

Take the square root of both sides to solve for rrr.r=324=18r = \sqrt{324} =18r=324​=18

STEP 9

The length of an arc is given by the formulaArc length=r×θArc\, length = r \times \thetaArclength=r×θ

STEP 10

Substitute the values of rrr and θ\thetaθ into the formula to find the length of arc UV.Arc length=18×π2Arc\, length =18 \times \frac{\pi}{2}Arclength=18×2π​

STEP 11

Calculate the length of arc UV.Arc length=18×π=9πArc\, length =18 \times \frac{\pi}{} =9 \piArclength=18×π​=9πThe length of arc UV is 9π9 \pi9π.

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QuestionFind the length of arc UV in a circle sector with area $81 \pi$ . Choices: (A) $6 \pi$ , (B) $9 \pi$ , (C) $27 \pi$ , (D) $54 \pi$ , (E) $108 \pi$ .

Assumptions1. The sector of the circle has an area of $81 \pi$ . . We need to find the length of arc UV.

The area of a sector of a circle is given by the formula $Area = \frac{r^2 \times \theta}{2}$ where $r$ is the radius of the circle and $\theta$ is the angle of the sector in radians.

We know that the area of the sector is $81 \pi$ . We can equate this to the formula for the area of a sector and solve for $r$ .
$81 \pi = \frac{r^2 \times \theta}{2}$

We can simplify the equation by multiplying both sides by2.
$162 \pi = r^2 \times \theta$

The angle $\theta$ for a full circle in radians is $2\pi$ . Since the sector is a quarter of a circle, $\theta$ is $\frac{2\pi}{4} = \frac{\pi}{2}$ .

Substitute $\theta$ into the equation and solve for $r$ .
$162 \pi = r^2 \times \frac{\pi}{2}$

implify the equation by multiplying both sides by $\frac{2}{\pi}$ .
$324 = r^2$

Take the square root of both sides to solve for $r$ .
$r = \sqrt{324} =18$

The length of an arc is given by the formula $Arc\, length = r \times \theta$

Substitute the values of $r$ and $\theta$ into the formula to find the length of arc UV.
$Arc\, length =18 \times \frac{\pi}{2}$

Calculate the length of arc UV.
$Arc\, length =18 \times \frac{\pi}{} =9 \pi$ The length of arc UV is $9 \pi$ .