Question

A dartboard consists of a circle inscribed in a square. The area of the circle is 25π square units. The area of the square is 100 square units.

A dartboard consists of a circle inscribed in a square.

Megan randomly throws a dart at the square. Assuming the dart lands within the square, what is the probability that the dart lands within the dartboard? Round your answer to the nearest tenth of a percent.

Answer 1

To find the probability that a dart thrown at a square lands within the inscribed circle (the dartboard), we can use the areas of the circle and the square.

1. **Area of the Circle**:
The area of the circle is given as:
\[
A_{\text{circle}} = 25\pi \text{ square units}
\]

2. **Area of the Square**:
The area of the square is given as:
\[
A_{\text{square}} = 100 \text{ square units}
\]

3. **Probability Calculation**:
The probability \( P \) that the dart lands within the circle is the ratio of the area of the circle to the area of the square:
\[
P = \frac{A_{\text{circle}}}{A_{\text{square}}}
\]

Substituting the areas:
\[
P = \frac{25\pi}{100}
\]

Simplifying this expression:
\[
P = \frac{25\pi}{100} = \frac{\pi}{4}
\]

4. **Numerical Value**:
To find the probability in numerical form, we approximate \( \pi \):
\[
P \approx \frac{3.14}{4} \approx 0.785
\]

5. **Percentage Conversion**:
Convert this probability into a percentage:
\[
P \times 100 \approx 0.785 \times 100 \approx 78.5\%
\]

6. **Rounding**:
Rounding to the nearest tenth of a percent gives us:
\[
\text{Probability} \approx 78.5\%
\]

Thus, the final answer is:
\[
\boxed{78.5\%}
\]