Question

Find the area of the shaded sector and express your answer as a fraction times pie

Answer 1

In order to find the length of the arc, first let's find the central angle corresponding to the arc.

To find it, let's use the formula for the area of a sector:

`A=(r^2\theta)/(2)`

Using A = 5pi and r = 6, we have:

`\begin{gathered} 5\pi=(6^2\theta)/(2)\\ \\ 36\theta=10\pi \\ \theta=(10)/(36)\pi \end{gathered}`

Now, to find the length of the arc, we have the formula below:

`\begin{gathered} l=\theta\cdot r\\ \\ l=(10\pi)/(36)\cdot6\\ \\ l=(10)/(6)\pi=(5)/(3)\pi \end{gathered}`

This is the smaller arc. To find the greater arc, we subtract the circumference by the smaller arc:

`\begin{gathered} arc=2\pi r-(5)/(3)\pi\\ \\ arc=12\pi-(5)/(3)\pi\\ \\ arc=(36)/(3)\pi-(5)/(3)\pi\\ \\ arc=(31)/(3)\pi \end{gathered}`

Therefore the arc is 31/3 pi.