Question 23 (1 point)
There is a balcony that forms part of a circle around a stage, and they need to put up a safety railing. How long of a railing do they need if the radius of the circle is 40 feet, and the arc takes up $45^{\circ}$ ? Use 3.14 for pi.

Step 1 ：The length of the railing needed is the length of the arc of the circle that the balcony takes up. The formula for the length of an arc of a circle is given by \(L = r \times \theta\), where \(r\) is the radius of the circle and \(\theta\) is the angle that the arc subtends at the center of the circle, measured in radians.

Step 2 ：In this case, the radius \(r\) is given as 40 feet and the angle \(\theta\) is given as \(45^\circ\). However, we need to convert this angle from degrees to radians before we can use it in the formula. The conversion factor is \(\frac{\pi}{180}\) radians per degree, so \(\theta = 45 \times \frac{\pi}{180}\) radians.

Step 3 ：Once we have the radius and the angle in the correct units, we can substitute them into the formula to find the length of the railing needed.

Step 4 ：Substituting the values into the formula, we get \(L = 40 \times 0.7853981633974483\), which simplifies to \(L = 31.41592653589793\).

Step 5 ：Final Answer: The length of the railing they need is approximately \(\boxed{31.42}\) feet.