Question
A circle with a radius of 4 inches and a square with a side length of 8 inches are both inscribed in a rectangle with a width of 19 inches and a length of 21 inches. The circle and square inscribed in the rectangle are beside each other. Determine the probability that a random point placed in the figure is not inside the circle or inside the square. Round your answer to the nearest hundredth.(1 point)
Responses
0.84
0.84
0.71
0.71
0.87
0.87
0.29
Responses
0.84
0.84
0.71
0.71
0.87
0.87
0.29
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To find the probability that a random point placed in the rectangle is not inside the circle or the square, we'll follow these steps:
1. **Calculate the area of the rectangle**:
\[
\text{Area}_{\text{rectangle}} = \text{length} \times \text{width} = 21 \, \text{inches} \times 19 \, \text{inches} = 399 \, \text{square inches}
\]
2. **Calculate the area of the circle**:
Given the radius \( r = 4 \) inches,
\[
\text{Area}_{\text{circle}} = \pi r^2 = \pi (4^2) = 16\pi \approx 50.27 \, \text{square inches} \quad (\text{using } \pi \approx 3.14)
\]
3. **Calculate the area of the square**:
The side length of the square is \( s = 8 \) inches,
\[
\text{Area}_{\text{square}} = s^2 = 8^2 = 64 \, \text{square inches}
\]
4. **Calculate the total area occupied by the circle and square**:
\[
\text{Area}_{\text{occupied}} = \text{Area}_{\text{circle}} + \text{Area}_{\text{square}} \approx 50.27 + 64 = 114.27 \, \text{square inches}
\]
5. **Calculate the area not occupied by the circle or square**:
\[
\text{Area}_{\text{not occupied}} = \text{Area}_{\text{rectangle}} - \text{Area}_{\text{occupied}} \approx 399 - 114.27 \approx 284.73 \, \text{square inches}
\]
6. **Calculate the probability that a random point is not inside the circle or the square**:
\[
\text{Probability} = \frac{\text{Area}_{\text{not occupied}}}{\text{Area}_{\text{rectangle}}} \approx \frac{284.73}{399} \approx 0.713
\]
7. **Round this value to the nearest hundredth**:
\[
0.713 \text{ rounds to } 0.71
\]
Thus, the probability that a random point placed in the figure is not inside the circle or inside the square is \( \boxed{0.71} \).
1. **Calculate the area of the rectangle**:
\[
\text{Area}_{\text{rectangle}} = \text{length} \times \text{width} = 21 \, \text{inches} \times 19 \, \text{inches} = 399 \, \text{square inches}
\]
2. **Calculate the area of the circle**:
Given the radius \( r = 4 \) inches,
\[
\text{Area}_{\text{circle}} = \pi r^2 = \pi (4^2) = 16\pi \approx 50.27 \, \text{square inches} \quad (\text{using } \pi \approx 3.14)
\]
3. **Calculate the area of the square**:
The side length of the square is \( s = 8 \) inches,
\[
\text{Area}_{\text{square}} = s^2 = 8^2 = 64 \, \text{square inches}
\]
4. **Calculate the total area occupied by the circle and square**:
\[
\text{Area}_{\text{occupied}} = \text{Area}_{\text{circle}} + \text{Area}_{\text{square}} \approx 50.27 + 64 = 114.27 \, \text{square inches}
\]
5. **Calculate the area not occupied by the circle or square**:
\[
\text{Area}_{\text{not occupied}} = \text{Area}_{\text{rectangle}} - \text{Area}_{\text{occupied}} \approx 399 - 114.27 \approx 284.73 \, \text{square inches}
\]
6. **Calculate the probability that a random point is not inside the circle or the square**:
\[
\text{Probability} = \frac{\text{Area}_{\text{not occupied}}}{\text{Area}_{\text{rectangle}}} \approx \frac{284.73}{399} \approx 0.713
\]
7. **Round this value to the nearest hundredth**:
\[
0.713 \text{ rounds to } 0.71
\]
Thus, the probability that a random point placed in the figure is not inside the circle or inside the square is \( \boxed{0.71} \).
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