Q.4. These sectors of circles are similar. Find the positive difference between the perimeters of the sectors.

• The arc length of the smaller sector is 6.
• The difference between the areas of the sectors is 21.

Answer Choices

• 4.5
• 7
• 8
• 9
• 10.5
• 14
• 15

Solution

Given that the arc length of the smaller sector is 6, we can find the angle $\theta$ as follows:

$$\theta = \frac{6}{r}$$

Using the formula for the area of a sector, the difference between the areas gives us the equation:

$$\frac{3(r + 3)^2}{r} - 3r = 21$$

Solving this equation gives $r = 9$.

The perimeters of the two sectors are:

• Smaller sector: $P_{\text{small}} = 24$
• Larger sector: $P_{\text{large}} = 32$

The positive difference between the perimeters is:

$$P_{\text{large}} - P_{\text{small}} = 32 - 24 = 8$$

Thus, the correct answer is: 8.