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 0.71 0.71 0.29 0.29 0.83 0.83 0.17 0.17
Answers
GPT-4o mini
To find the probability that a randomly placed point in the rectangle is inside the square, we will first calculate the areas of the rectangle, the square, and ultimately find the probability based on these areas.
1. **Area of the rectangle**:
The rectangle has a width of 12 inches and a length of 8 inches.
\[\text{Area of rectangle} = \text{width} \times \text{length} = 12 \times 8 = 96 \text{ square inches}\]
2. **Area of the square**:
The square has a side length of 4 inches.
\[\text{Area of square} = \text{side}^2 = 4^2 = 16 \text{ square inches}\]
3. **Probability of a point being inside the square**:
The probability \(P\) that a point chosen randomly in the rectangle falls inside the square is given by the ratio of the area of the square to the area of the rectangle:
\[
P = \frac{\text{Area of square}}{\text{Area of rectangle}} = \frac{16}{96} = \frac{1}{6} \approx 0.1667
\]
4. **Rounding to the nearest hundredth**:
When rounded to the nearest hundredth, the probability \(P\) is approximately 0.17.
Thus, the final answer is \( \boxed{0.17} \).
1. **Area of the rectangle**:
The rectangle has a width of 12 inches and a length of 8 inches.
\[\text{Area of rectangle} = \text{width} \times \text{length} = 12 \times 8 = 96 \text{ square inches}\]
2. **Area of the square**:
The square has a side length of 4 inches.
\[\text{Area of square} = \text{side}^2 = 4^2 = 16 \text{ square inches}\]
3. **Probability of a point being inside the square**:
The probability \(P\) that a point chosen randomly in the rectangle falls inside the square is given by the ratio of the area of the square to the area of the rectangle:
\[
P = \frac{\text{Area of square}}{\text{Area of rectangle}} = \frac{16}{96} = \frac{1}{6} \approx 0.1667
\]
4. **Rounding to the nearest hundredth**:
When rounded to the nearest hundredth, the probability \(P\) is approximately 0.17.
Thus, the final answer is \( \boxed{0.17} \).