Question

A circle with a radius of 3 inches and a square with a side length of 4 inches are both inscribed in a rectangle with a width of 12 inches and a length of 8 inches. The circle and square are next to each other in the rectangle. Determine the probability that a point placed randomly in the figure is inside the square. Round your answer to the nearest hundredth.(1 point)
Responses

0.29
0.29

0.83
0.83

0.71
0.71

0.17

Answer 1

To find the probability that a randomly placed point in the rectangle is inside the square, we will follow these steps:

1. **Calculate the area of the rectangle.**
\[
\text{Area of rectangle} = \text{length} \times \text{width} = 8\, \text{inches} \times 12\, \text{inches} = 96\, \text{square inches}
\]

2. **Calculate the area of the square.**
\[
\text{Area of square} = \text{side}^2 = 4\, \text{inches} \times 4\, \text{inches} = 16\, \text{square inches}
\]

3. **Calculate the probability that a point is in the square.**
The probability \( P \) that a point randomly chosen in the rectangle lies within the square is given by the ratio of the area of the square to the area of the rectangle:
\[
P = \frac{\text{Area of square}}{\text{Area of rectangle}} = \frac{16}{96}
\]

4. **Simplify the fraction.**
\[
P = \frac{16}{96} = \frac{1}{6} \approx 0.1667
\]

5. **Round to the nearest hundredth.**
\[
P \approx 0.17
\]

Thus, the probability that a point placed randomly in the figure is inside the square is \( \boxed{0.17} \).