Answer:
\(\[\boxed{\frac{7}{3} \pi \, \text{cm}^2}\]\)
Step-by-step explanation:
1. Find the area of the sector \(\( AOB \)\) of the larger circle:
The larger circle has a radius of 4 cm and the sector \(\( AOB \)\) subtends an angle of \(\( 120^\circ \)\).
\(\[ \text{Area of sector } AOB = \frac{120^\circ}{360^\circ} \times \pi \times (4 \, \text{cm})^2 = \frac{1}{3} \pi \times 16 \, \text{cm}^2 = \frac{16}{3} \pi \, \text{cm}^2 \]\)
2. Find the area of the sector \(\( DOB \)\) of the smaller circle:
The smaller circle has a radius of 3 cm and the sector \( DOB \) also subtends an angle of \(\( 120^\circ \)\).
\(\[ \text{Area of sector } DOB = \frac{120^\circ}{360^\circ} \times \pi \times (3 \, \text{cm})^2 = \frac{1}{3} \pi \times 9 \, \text{cm}^2 = 3 \pi \, \text{cm}^2 \]\)
3. Find the area of the shaded region:
The shaded region is the difference between the area of sector \(\( AOB \)\) and the area of sector\(\( DOB \)\).
\(\[ \text{Area of shaded region} = \frac{16}{3} \pi \, \text{cm}^2 - 3 \pi \, \text{cm}^2 \]\)
Simplify the expression:
\(\[ \frac{16}{3} \pi - 3 \pi = \frac{16}{3} \pi - \frac{9}{3} \pi = \frac{16 - 9}{3} \pi = \frac{7}{3} \pi \]\)
Therefore, the area of the shaded regions is:
\(\[\boxed{\frac{7}{3} \pi \, \text{cm}^2}\]\)
