Math Snap

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.
$$Arc\, length =18 \times \frac{\pi}{2}$$

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