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)

Question

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 \( \boxed{0.17} \).