Answer:
[tex]\huge\boxed{15 \pi \ \text{or} \approx 47.1 \ \text{in.}}[/tex]
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.
First step: Find the circumference of the circle.
The circumference of any circle can be defined by the formula [tex]2 \pi r[/tex], where r is the radius of the circle. The radius is given to us, 30 in. We can now substitute that into the formula.
- [tex]2\cdot \pi \cdot 30[/tex]
- [tex]60 \cdot \pi[/tex]
- [tex]60\pi[/tex]
So our circumference is 60π.
Second Step: Find the ratio of the central angle of the arc to the total circle degrees
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.
[tex]\frac{90}{360}[/tex]
Third Step: Equal out the two proportions and solve for the missing arc length
Now that we have our base proportion ([tex]\frac{90}{360}[/tex]), we can turn 60π into a proportion as well, leaving 60π as the denominator so we can solve for the arc length.
[tex]\frac{x}{60 \pi} = \frac{90}{360}[/tex]
We can now solve for x by cross multiplying.
- [tex]\frac{90}{360} = \frac{1}{4}[/tex]
- [tex]\frac{x}{60\pi} = \frac{1}{4}[/tex]
- [tex]x = \frac{60\pi \cdot 1}{4}[/tex]
- [tex]x = \frac{60\pi}{4}[/tex]
- [tex]x = 15\pi \approx47.1[/tex]
Hope this helped!
