a sector has radius 4 in and area $\frac{296}{45}pi$ in$^2$. what is the measure, in degrees, of the central…

a sector has radius 4 in and area $\frac{296}{45}pi$ in$^2$. what is the measure, in degrees, of the central angle of the sector?\nshow your work here\nhint: to add an exponent ($x^y$), type \exponent\ or press \^\

a sector has radius 4 in and area $\frac{296}{45}pi$ in$^2$. what is the measure, in degrees, of the central angle of the sector?\nshow your work here\nhint: to add an exponent ($x^y$), type \exponent\ or press \^\

Answer

Explanation:

Step1: Recall sector area formula

The formula for the area of a sector is $A = \frac{\theta}{360^\circ} \times \pi r^2$, where $\theta$ is the central angle in degrees, and $r$ is the radius.

Step2: Substitute given values

Given $A = \frac{296}{45}\pi$ and $r = 4$, substitute into the formula: $\frac{296}{45}\pi = \frac{\theta}{360^\circ} \times \pi (4)^2$ Simplify $(4)^2 = 16$: $\frac{296}{45}\pi = \frac{\theta}{360^\circ} \times 16\pi$

Step3: Cancel $\pi$ from both sides

$\frac{296}{45} = \frac{16\theta}{360^\circ}$ Simplify $\frac{16}{360^\circ} = \frac{1}{22.5^\circ}$: $\frac{296}{45} = \frac{\theta}{22.5^\circ}$

Step4: Solve for $\theta$

Multiply both sides by $22.5^\circ$: $\theta = \frac{296}{45} \times 22.5^\circ$ Note that $22.5^\circ = \frac{45}{2}^\circ$, so: $\theta = \frac{296}{45} \times \frac{45}{2}^\circ = 148^\circ$

Answer:

$148^\circ$