We are asked to find the length of an arc that has a $90$ degree central angle in a circle with a radius of $15$. The answer should be rounded to the nearest hundredth.

1. Problem Description

We are asked to find the length of an arc that has a $90$ degree central angle in a circle with a radius of $15$. The answer should be rounded to the nearest hundredth.

2. Solution Steps

The formula for the arc length $s$ of a circle with radius $r$ and central angle $\theta$ (in radians) is given by:

$s = r\theta$

First, we need to convert the central angle from degrees to radians. We know that $180$ degrees is equal to $\pi$ radians. Therefore,

$\theta \text{ (radians)} = \frac{\theta \text{ (degrees)}}{180} \times \pi$

In our case, the central angle is $90$ degrees, so

$\theta = \frac{90}{180} \pi = \frac{1}{2} \pi = \frac{\pi}{2}$ radians.

The radius of the circle is given as $r = 15$.

Now, we can use the arc length formula:

$s = r\theta = 15 \times \frac{\pi}{2} = \frac{15\pi}{2}$

Using a calculator, we can approximate this value:

$s \approx \frac{15 \times 3.14159}{2} \approx \frac{47.12385}{2} \approx 23.561925$

Rounding to the nearest hundredth, we have:

$s \approx 23.56$

3. Final Answer

23.56