Question

a circle has an arc of length 23 with central angle 8°. find the circumference of the circle.

Explanation:

Step1: Recall the arc - length formula

The formula for the length of an arc \(s\) is \(s=\frac{\theta}{360^{\circ}}\times C\), where \(\theta\) is the central angle in degrees and \(C\) is the circumference of the circle.

Step2: Substitute the given values into the formula

We are given that \(s = 23\) and \(\theta=8^{\circ}\). Substituting these into the formula \(s=\frac{\theta}{360^{\circ}}\times C\), we get \(23=\frac{8^{\circ}}{360^{\circ}}\times C\).

Step3: Solve for \(C\)

First, simplify the fraction \(\frac{8}{360}=\frac{2}{90}=\frac{1}{45}\). Then our equation is \(23=\frac{1}{45}C\).
Multiply both sides of the equation by \(45\) to isolate \(C\): \(C = 23\times45\).
Calculate \(23\times45=(20 + 3)\times45=20\times45+3\times45=900+135 = 1035\).

Answer:

$1035$