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 will be inside the circle? Round your answer to the nearest hundredth.(1 point)

Answer 1

To find the probability that a point placed randomly in the figure (the square) will be inside the circle, we need to calculate the areas of both the circle and the square.

1. **Area of the circle**:
The formula for the area \(A\) of a circle is given by:

\[
A = \pi r^2
\]

where \(r\) is the radius. Here, the radius \(r = 4\) cm.

\[
A_{\text{circle}} = \pi (4^2) = \pi \times 16 = 16\pi
\]

2. **Area of the square**:
The formula for the area \(A\) of a square is given by:

\[
A = s^2
\]

where \(s\) is the side length. Here, the side length \(s = 8\) cm.

\[
A_{\text{square}} = 8^2 = 64
\]

3. **Probability of selecting a point inside the circle**:
The probability \(P\) is given by the ratio of the area of the circle to the area of the square.

\[
P = \frac{A_{\text{circle}}}{A_{\text{square}}} = \frac{16\pi}{64} = \frac{\pi}{4}
\]

Using the approximate value of \(\pi \approx 3.14\):

\[
P \approx \frac{3.14}{4} \approx 0.785
\]

4. **Rounding to the nearest hundredth**:
We round \(0.785\) to:

\[
0.79
\]

Thus, the probability that a randomly placed point in the square will be inside the circle is:

\[
\boxed{0.79}
\]