find radius of circle given sector area and angle

1. Identify the formula for the area of a sector: The area of a sector of a circle is given by the formula $A = \frac{\theta}{360^\circ} \cdot \pi r^2$, where $\theta$ is the central angle in degrees and $r$ is the radius of the circle.

2. Substitute the known values into the formula: We are given that the area of the shaded sector is $15\pi$ and the central angle $\theta$ is $150^\circ$. Plugging these into the formula, we get:
$$15\pi = \frac{150^\circ}{360^\circ} \cdot \pi r^2$$

3. Simplify the equation: First, simplify the fraction $\frac{150}{360}$:
$$\frac{150}{360} = \frac{15}{36} = \frac{5}{12}$$
So the equation becomes:
$$15\pi = \frac{5}{12} \cdot \pi r^2$$

4. Solve for $r^2$: Divide both sides by $\pi$ and multiply by the reciprocal of $\frac{5}{12}$ (which is $\frac{12}{5}$):
$$15 = \frac{5}{12} r^2$$
$$15 \cdot \frac{12}{5} = r^2$$
$$3 \cdot 12 = r^2$$
$$36 = r^2$$

5. Find the radius $r$: Take the square root of both sides. Since the radius must be a positive value:
$$r = \sqrt{36} = 6$$

The radius of circle $S$ is 6 feet.