Question

the area of the shaded - sector is 15\pi. the area of \\(\odot p\\) is 150\pi. what is the value of \\(x\\)?
a. 10
b. 15
c. 18
d. 30
e. 36

Explanation:

Step1: Recall sector - area formula

The formula for the area of a sector of a circle is $A=\frac{\theta}{360^{\circ}}\times\pi r^{2}$, where $A$ is the area of the sector, $\theta$ is the central - angle of the sector in degrees, and $\pi r^{2}$ is the area of the whole circle. Let the central - angle of the shaded sector be $\theta$ and the area of the whole circle be $A_{circle}$ and the area of the sector be $A_{sector}$. We know that $A_{sector} = 15\pi$ and $A_{circle}=150\pi$.

Step2: Substitute values into formula

Substitute the given values into the formula $\frac{A_{sector}}{A_{circle}}=\frac{\theta}{360^{\circ}}$. So, $\frac{15\pi}{150\pi}=\frac{\theta}{360^{\circ}}$.

Step3: Simplify the left - hand side

Simplify $\frac{15\pi}{150\pi}$ to get $\frac{15}{150}=\frac{1}{10}$. So, $\frac{1}{10}=\frac{\theta}{360^{\circ}}$.

Step4: Solve for $\theta$

Cross - multiply: $10\theta = 360^{\circ}$, then $\theta = 36^{\circ}$.

Answer:

E. 36