Question

18
select the correct answer.
what is the area of the shaded portion of the figure? express your answer in terms of pi.
ae = 4 cm
ab = 6 cm
a. \\(\frac{56}{9}\pi\\)
b. \\(\frac{728}{45}\pi\\)
c. \\(\frac{56}{5}\pi\\)
d. \\(\frac{224}{45}\pi\\)

Explanation:

Step1: Recall sector area formula

The area of a sector is $\frac{\theta}{360^\circ} \times \pi r^2$, where $\theta$ is the central angle, $r$ is the radius.

Step2: Calculate area of large sector

Large radius $AB=6$ cm, $\theta=112^\circ$.
$\text{Area of large sector} = \frac{112^\circ}{360^\circ} \times \pi \times 6^2 = \frac{112}{360} \times 36\pi = \frac{112}{10}\pi = \frac{56}{5}\pi$

Step3: Calculate area of small sector

Small radius $AE=4$ cm, $\theta=112^\circ$.
$\text{Area of small sector} = \frac{112^\circ}{360^\circ} \times \pi \times 4^2 = \frac{112}{360} \times 16\pi = \frac{1792}{360}\pi = \frac{224}{45}\pi$

Step4: Find shaded area (difference)

Subtract small sector area from large sector area.
$\text{Shaded Area} = \frac{56}{5}\pi - \frac{224}{45}\pi = \frac{504}{45}\pi - \frac{224}{45}\pi = \frac{280}{45}\pi = \frac{56}{9}\pi$

Answer:

A. $\frac{56}{9}\pi$