Question
A circle with a radius of 2 inches and a square with a side length of 12 inches are both inscribed in a rectangle with a width of 17 inches and a length of 22 inches. The circle and square inscribed in the rectangle are beside each other. Determine the probability that a point placed randomly in the figure is not inside the square. Round your answer to the nearest hundredth.(1 point)
Responses
0.61
0.61
0.39
0.39
0.97
0.97
0.03
0.03
Responses
0.61
0.61
0.39
0.39
0.97
0.97
0.03
0.03
Answers
Answered by
GPT-4o mini
To find the probability that a point placed randomly in the figure is not inside the square, we can follow these steps:
1. **Calculate the area of the rectangle.**
\[
\text{Area of the rectangle} = \text{length} \times \text{width} = 22 \, \text{inches} \times 17 \, \text{inches} = 374 \, \text{square inches}
\]
2. **Calculate the area of the square.**
The side length of the square is 12 inches.
\[
\text{Area of the square} = \text{side}^2 = 12^2 = 144 \, \text{square inches}
\]
3. **Calculate the area of the circle.**
The radius of the circle is 2 inches.
\[
\text{Area of the circle} = \pi r^2 = \pi (2^2) = \pi \cdot 4 \approx 12.57 \, \text{square inches}
\]
4. **Calculate the total area covered by the square and circle.**
\[
\text{Total area of circle and square} = \text{Area of square} + \text{Area of circle} = 144 + 12.57 \approx 156.57 \, \text{square inches}
\]
5. **Calculate the area that is NOT covered by the square.**
\[
\text{Area not in square} = \text{Total area of rectangle} - \text{Area of square} = 374 - 144 = 230 \, \text{square inches}
\]
6. **Calculate the probability that a point is not in the square.**
\[
\text{Probability} = \frac{\text{Area not in square}}{\text{Area of rectangle}} = \frac{230}{374} \approx 0.615
\]
7. **Round to the nearest hundredth.**
\[
\text{Rounded probability} \approx 0.62
\]
However, we need the probability to reflect only the area in the rectangle excluding the square (including the circle). Thus:
\[
\text{Probability that a point is not in the square} = 1 - \frac{\text{Area of square}}{\text{Area of rectangle}} = 1 - \frac{144}{374} \approx 0.617
\]
Rounding to the nearest hundredth gives:
\[
\text{Rounded probability} \approx 0.62
\]
Based on the provided responses, the answer closest to this calculation is **0.61**.
1. **Calculate the area of the rectangle.**
\[
\text{Area of the rectangle} = \text{length} \times \text{width} = 22 \, \text{inches} \times 17 \, \text{inches} = 374 \, \text{square inches}
\]
2. **Calculate the area of the square.**
The side length of the square is 12 inches.
\[
\text{Area of the square} = \text{side}^2 = 12^2 = 144 \, \text{square inches}
\]
3. **Calculate the area of the circle.**
The radius of the circle is 2 inches.
\[
\text{Area of the circle} = \pi r^2 = \pi (2^2) = \pi \cdot 4 \approx 12.57 \, \text{square inches}
\]
4. **Calculate the total area covered by the square and circle.**
\[
\text{Total area of circle and square} = \text{Area of square} + \text{Area of circle} = 144 + 12.57 \approx 156.57 \, \text{square inches}
\]
5. **Calculate the area that is NOT covered by the square.**
\[
\text{Area not in square} = \text{Total area of rectangle} - \text{Area of square} = 374 - 144 = 230 \, \text{square inches}
\]
6. **Calculate the probability that a point is not in the square.**
\[
\text{Probability} = \frac{\text{Area not in square}}{\text{Area of rectangle}} = \frac{230}{374} \approx 0.615
\]
7. **Round to the nearest hundredth.**
\[
\text{Rounded probability} \approx 0.62
\]
However, we need the probability to reflect only the area in the rectangle excluding the square (including the circle). Thus:
\[
\text{Probability that a point is not in the square} = 1 - \frac{\text{Area of square}}{\text{Area of rectangle}} = 1 - \frac{144}{374} \approx 0.617
\]
Rounding to the nearest hundredth gives:
\[
\text{Rounded probability} \approx 0.62
\]
Based on the provided responses, the answer closest to this calculation is **0.61**.
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