Question

A circle with a radius of 4 centimeters is inscribed in a square with a side length of 8 centimeters. What is the probability that a point placed randomly in the figure is not inside the circle? Round your answer to the nearest hundredth.(1 point)

Answer 1

To solve the problem, we first need to find the areas of both the square and the circle.

1. **Area of the square:**
The area \( A_{\text{square}} \) of a square is given by the formula:
\[
A_{\text{square}} = \text{side}^2
\]
Here, the side length of the square is 8 cm, so:
\[
A_{\text{square}} = 8^2 = 64 \text{ cm}^2
\]

2. **Area of the circle:**
The area \( A_{\text{circle}} \) of a circle is given by the formula:
\[
A_{\text{circle}} = \pi r^2
\]
Here, the radius \( r \) of the circle is 4 cm, so:
\[
A_{\text{circle}} = \pi (4^2) = \pi \cdot 16 = 16\pi \text{ cm}^2
\]

3. **Area not inside the circle:**
The area outside the circle but inside the square is given by:
\[
A_{\text{outside}} = A_{\text{square}} - A_{\text{circle}} = 64 - 16\pi
\]

4. **Probability that a point placed randomly is not inside the circle:**
The probability \( P \) that a point chosen randomly from the square is outside the circle is:
\[
P = \frac{A_{\text{outside}}}{A_{\text{square}}} = \frac{64 - 16\pi}{64}
\]

5. **Calculating the probability:**
We will compute this probability and round it to the nearest hundredth. Using \( \pi \approx 3.14 \), we can calculate:
\[
A_{\text{circle}} \approx 16 \cdot 3.14 = 50.24
\]
\[
A_{\text{outside}} \approx 64 - 50.24 = 13.76
\]
\[
P \approx \frac{13.76}{64} \approx 0.215
\]

Now, rounding to the nearest hundredth:
\[
P \approx 0.22
\]

Thus, the probability that a point placed randomly in the figure is not inside the circle is \( \boxed{0.22} \).