Question
A circle with a radius of 6 inches and a square with a side length of 10 inches are both inscribed in a rectangle with a width of 24 inches and a length of 30 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 circle. Round your answer to the nearest hundredth.(1 point) Responses 0.86 0.86 0.14 0.14 0.16 0.16 0.84
Answers
GPT-4o mini
To determine the probability that a randomly placed point in the rectangle is not inside the circle, we need to calculate the areas of the rectangle, circle, and square.
1. **Area of the Rectangle**:
\[
\text{Area}_{\text{rectangle}} = \text{length} \times \text{width} = 30 \, \text{inches} \times 24 \, \text{inches} = 720 \, \text{square inches}
\]
2. **Area of the Circle**:
The formula for the area of a circle is:
\[
\text{Area}_{\text{circle}} = \pi r^2
\]
where \( r = 6 \, \text{inches} \):
\[
\text{Area}_{\text{circle}} = \pi (6^2) = \pi \times 36 \approx 113.10 \, \text{square inches} \quad (\text{using } \pi \approx 3.14)
\]
3. **Area of the Square**:
The formula for the area of a square is:
\[
\text{Area}_{\text{square}} = \text{side length}^2 = 10^2 = 100 \, \text{square inches}
\]
4. **Total Area of Circle and Square**:
\[
\text{Total Area}_{\text{circle and square}} = \text{Area}_{\text{circle}} + \text{Area}_{\text{square}} \approx 113.10 + 100 = 213.10 \, \text{square inches}
\]
5. **Area Not Inside the Circle**:
To find the area not inside the circle, we can subtract the area of the circle from the total area of the rectangle:
\[
\text{Area not inside the circle} = \text{Area}_{\text{rectangle}} - \text{Area}_{\text{circle}} \approx 720 - 113.10 \approx 606.90 \, \text{square inches}
\]
6. **Probability that a Point is Not Inside the Circle**:
The probability that a randomly selected point in the rectangle is not inside the circle is given by:
\[
P(\text{not in circle}) = \frac{\text{Area not inside the circle}}{\text{Area}_{\text{rectangle}}}
\]
\[
P(\text{not in circle}) = \frac{606.90}{720} \approx 0.842
\]
Rounded to the nearest hundredth, this value is \( 0.84 \).
Thus, the final answer is:
\[
\boxed{0.84}
\]
1. **Area of the Rectangle**:
\[
\text{Area}_{\text{rectangle}} = \text{length} \times \text{width} = 30 \, \text{inches} \times 24 \, \text{inches} = 720 \, \text{square inches}
\]
2. **Area of the Circle**:
The formula for the area of a circle is:
\[
\text{Area}_{\text{circle}} = \pi r^2
\]
where \( r = 6 \, \text{inches} \):
\[
\text{Area}_{\text{circle}} = \pi (6^2) = \pi \times 36 \approx 113.10 \, \text{square inches} \quad (\text{using } \pi \approx 3.14)
\]
3. **Area of the Square**:
The formula for the area of a square is:
\[
\text{Area}_{\text{square}} = \text{side length}^2 = 10^2 = 100 \, \text{square inches}
\]
4. **Total Area of Circle and Square**:
\[
\text{Total Area}_{\text{circle and square}} = \text{Area}_{\text{circle}} + \text{Area}_{\text{square}} \approx 113.10 + 100 = 213.10 \, \text{square inches}
\]
5. **Area Not Inside the Circle**:
To find the area not inside the circle, we can subtract the area of the circle from the total area of the rectangle:
\[
\text{Area not inside the circle} = \text{Area}_{\text{rectangle}} - \text{Area}_{\text{circle}} \approx 720 - 113.10 \approx 606.90 \, \text{square inches}
\]
6. **Probability that a Point is Not Inside the Circle**:
The probability that a randomly selected point in the rectangle is not inside the circle is given by:
\[
P(\text{not in circle}) = \frac{\text{Area not inside the circle}}{\text{Area}_{\text{rectangle}}}
\]
\[
P(\text{not in circle}) = \frac{606.90}{720} \approx 0.842
\]
Rounded to the nearest hundredth, this value is \( 0.84 \).
Thus, the final answer is:
\[
\boxed{0.84}
\]