Answer:
![\huge\boxed{15 \pi \ \text{or} \approx 47.1 \ \text{in.}}](https://tex.z-dn.net/?f=%5Chuge%5Cboxed%7B15%20%5Cpi%20%5C%20%5Ctext%7Bor%7D%20%5Capprox%2047.1%20%5C%20%5Ctext%7Bin.%7D%7D)
Step-by-step explanation:
We can note a couple of relationships in this circle.
The arc length will be a fraction of the circumference. It will be the same fraction of the circumference that the central angle is to the entire circle.
<u>First step: Find the circumference of the circle</u><u>.</u>
The circumference of any circle can be defined by the formula
, where r is the radius of the circle. The radius is given to us, 30 in. We can now substitute that into the formula.
So our circumference is 60π.
<u>Second Step: Find the ratio of the central angle of the arc to the total circle degrees</u>
We know that the total amount of degrees in a circle is 360°. Therefore, we can set up a proportion to find the ratio between the central angle (90°) and the total circle measurement.
![\frac{90}{360}](https://tex.z-dn.net/?f=%5Cfrac%7B90%7D%7B360%7D)
<u>Third Step: Equal out the two proportions and solve for the missing arc length</u>
Now that we have our base proportion (
), we can turn 60π into a proportion as well, leaving 60π as the denominator so we can solve for the arc length.
![\frac{x}{60 \pi} = \frac{90}{360}](https://tex.z-dn.net/?f=%5Cfrac%7Bx%7D%7B60%20%5Cpi%7D%20%3D%20%5Cfrac%7B90%7D%7B360%7D)
We can now solve for x by cross multiplying.
Hope this helped!