Answer
$240^{\circ}$
Work Step by Step
Step 1: The formula to be used here is $A=\frac{1}{2}r^{2}\theta$ where $A$ is the area of the sector of the circle of radius $r$ and central angle $\theta$.
Step 2: Substituting the values in the question, $96\pi=\frac{1}{2}(12^{2})\theta$
Step 3: Multiplying both sides by 2,
$96\pi\times2=\frac{1}{2}\times(12^{2})\times\theta\times2$
Step 4: $192\pi=144\theta$
Step 5: Multiplying both sides by $\frac{1}{144}$,
$192\pi\times\frac{1}{144}=144\theta\times\frac{1}{144}$
Step 6: $\theta=\frac{4\pi}{3}$
Step 7: Converting radians to degrees,
$\frac{4\pi}{3}=\frac{4\pi}{3}(\frac{180^{\circ}}{\pi})=240^{\circ}$
Step 8: Therefore, the required angle is $240^{\circ}$.