Question

find the central angle θ which forms a sector of area 9 square feet of a circle of radius 1.2 feet

Explanation:

Step1: Recall the formula for the area of a sector

The formula for the area of a sector of a circle is \(A=\frac{1}{2}r^{2}\theta\), where \(A\) is the area of the sector, \(r\) is the radius of the circle, and \(\theta\) is the central angle in radians.

Step2: Substitute the given values into the formula

We are given that \(A = 9\) square feet and \(r=1.2\) feet. Substituting these values into the formula \(A=\frac{1}{2}r^{2}\theta\), we get \(9=\frac{1}{2}(1.2)^{2}\theta\).
First, calculate \((1.2)^{2}=1.44\). Then the equation becomes \(9=\frac{1}{2}\times1.44\theta\).
Since \(\frac{1}{2}\times1.44 = 0.72\), the equation is \(9 = 0.72\theta\).

Step3: Solve for \(\theta\)

To solve for \(\theta\), divide both sides of the equation \(9 = 0.72\theta\) by \(0.72\). So \(\theta=\frac{9}{0.72}\).
\(\frac{9}{0.72}=\frac{900}{72}=\frac{900\div 36}{72\div 36}=\frac{25}{2}=12.5\) radians.

Answer:

The central angle \(\theta\) is \(12.5\) radians.