Question

what is the length of an arc if we know the radius of curvature is 12 in. and the area of the sector created is 252 in²?
a) 36 in
b) 42 in
c) 12 in
d) 24 in

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector is \(A=\frac{1}{2}r\times l\), where \(A\) is the area of the sector, \(r\) is the radius, and \(l\) is the arc - length.

Step2: Substitute the given values into the formula

We are given that \(A = 252\space in^{2}\) and \(r=12\space in\). Substituting these values into the formula \(A=\frac{1}{2}r\times l\), we get \(252=\frac{1}{2}\times12\times l\).

Step3: Solve for \(l\)

First, simplify the right - hand side of the equation: \(\frac{1}{2}\times12\times l = 6l\). So the equation becomes \(6l=252\). Then, divide both sides of the equation by 6: \(l=\frac{252}{6}\).

Answer:

\(l = 42\space in\), so the answer is B.