- Calculate the area of the circle using the formula \( A = \pi r^2 \), where \( r = 25 \) mi:
\[
A_{\text{circle}} = \pi \times 25^2 = 625\pi \text{ mi}^2
\]
- Calculate the area of the trapezoid using the formula \( A = \frac{1}{2} \times (b_1 + b_2) \times h \), where \( b_1 = 8 \) mi, \( b_2 = 15 \) mi, and \( h = 30 \) mi:
\[
A_{\text{trapezoid}} = \frac{1}{2} \times (8 + 15) \times 30 = \frac{1}{2} \times 23 \times 30 = 345 \text{ mi}^2
\]
- Calculate the area of the shaded region by subtracting the area of the trapezoid from the area of the circle:
\[
A_{\text{shaded}} = 625\pi - 345 \text{ mi}^2
\]
- Calculate the probability of choosing a point in the shaded region by dividing the shaded area by the total area of the circle:
\[
P = \frac{625\pi - 345}{625\pi}
\]
- Simplify the expression:
\[
P = 1 - \frac{345}{625\pi}
\]
- Approximate the value:
\[
P \approx 1 - \frac{345}{625 \times 3.1416} \approx 1 - \frac{345}{1963.5} \approx 1 - 0.1757 \approx 0.8243
\]
- The probability is approximately \( 0.824 \) or \( 82.4\% \).
Parent Tip: Review the logic above to help your child master the concept of shaded area problems geometric probability worksheet.
