To solve this problem, we need to calculate the area of the fan's fabric, which forms a sector of a circle with a central angle of $160^\circ$ and a radius of 10.5 cm.

Step-by-step solution:

1. Formula for the area of a sector: The area $A$ of a sector of a circle is given by the formula: $A = \frac{\theta}{360^\circ} \times \pi r^2$ where:

   • $\theta = 160^\circ$ is the central angle of the sector,
   • $r = 10.5 \, \text{cm}$ is the radius of the fan.

2. Substitute the values: $A = \frac{160^\circ}{360^\circ} \times \pi (10.5)^2$ Simplifying the fraction: $A = \frac{4}{9} \times \pi (10.5)^2$ Calculate $(10.5)^2$: $10.5^2 = 110.25$ Now substitute this into the formula: $A = \frac{4}{9} \times \pi \times 110.25$

3. Approximate calculation: Using $\pi \approx 3.1416$: $A \approx \frac{4}{9} \times 3.1416 \times 110.25$ First, calculate $3.1416 \times 110.25$: $3.1416 \times 110.25 \approx 346.36$ Now, multiply by $\frac{4}{9}$: $A \approx \frac{4}{9} \times 346.36 \approx 153.94 \, \text{cm}^2$

Conclusion:

The area of the fabric on the fan is approximately $153.94 \, \text{cm}^2$.

Would you like more details or have any further questions?

Related questions:

1. How would the area change if the central angle were 180° instead of 160°?
2. What is the length of the arc formed by the fan's fabric?
3. How would the area change if the radius were doubled?
4. Can you find the perimeter of the sector, including the two radii and the arc?
5. How would you calculate the fabric area if the fan were partially folded?

Tip:

Always ensure the angle is in degrees when applying the sector area formula for simple calculations.