This problem involves the formula for the area of a sector of a circle. The area $A$ of a sector is given by the formula:

$A = \frac{1}{2} r^2 \theta$

where:

• $r$ is the radius of the circle,
• $\theta$ is the angle of the sector in radians,
• $A$ is the area of the sector.

Part (a):

In the image, for part (a), the sector's angle $\theta$ is $0.6 \, \text{rad}$ and the area of the sector is given as 28 square units.

We can solve for the radius $r$ using the formula:

$28 = \frac{1}{2} r^2 (0.6)$

Now, solve for $r$:

1. Multiply both sides by 2 to get rid of the fraction: $56 = r^2 \cdot 0.6$

2. Divide both sides by $0.6$: $r^2 = \frac{56}{0.6} = 93.33$

3. Take the square root of both sides: $r = \sqrt{93.33} \approx 9.66$

So, the radius $r$ for part (a) is approximately 9.66 units.

Part (b):

In part (b), the angle $\theta$ is given in degrees (110°), so we first need to convert this to radians. The conversion from degrees to radians is:

$\theta = \frac{110^\circ \times \pi}{180^\circ} = \frac{11\pi}{18} \approx 1.91986 \, \text{rad}$

Now, using the same formula:

$28 = \frac{1}{2} r^2 (1.91986)$

Solve for $r$:

1. Multiply both sides by 2: $56 = r^2 \cdot 1.91986$

2. Divide both sides by $1.91986$: $r^2 = \frac{56}{1.91986} \approx 29.17$

3. Take the square root: $r = \sqrt{29.17} \approx 5.4$

So, the radius $r$ for part (b) is approximately 5.4 units.

Would you like further explanation or details on the steps?

Here are 5 related questions:

1. What happens to the area if the angle $\theta$ is doubled while keeping the radius constant?
2. How do you convert radians back to degrees?
3. How does the area change if the radius is halved?
4. Can the formula for the sector area be used for non-circular shapes?
5. What other formulas use radians for calculating areas or arc lengths?

Tip: Always check if the angle is given in degrees or radians before applying formulas involving angles.