Question

given a circle of radius 14 km, find the length of the arc subtended by a central angle of 315°. round decimals to the nearest hundredth.

Explanation:

Step1: Convert angle to radians

First, convert $315^{\circ}$ to radians. We know that $1^{\circ}=\frac{\pi}{180}$ radians. So, $315^{\circ}=315\times\frac{\pi}{180}=\frac{7\pi}{4}$ radians.

Step2: Use arc - length formula

The formula for the length of an arc $s$ of a circle with radius $r$ and central angle $\theta$ (in radians) is $s = r\theta$. Here, $r = 14$ km and $\theta=\frac{7\pi}{4}$. So, $s=14\times\frac{7\pi}{4}=\frac{98\pi}{4}=\frac{49\pi}{2}$ km.

Step3: Calculate the numerical value

Now, calculate $\frac{49\pi}{2}$. $\frac{49\pi}{2}=\frac{49\times3.14159}{2}\approx76.97$ km.

Answer:

$76.97$ km