To determine the probability that a randomly chosen point in the rectangle is not inside the circle, we follow these steps:
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Calculate the area of the rectangle: \[ \text{Area of rectangle} = \text{length} \times \text{width} = 30 , \text{inches} \times 24 , \text{inches} = 720 , \text{square inches} \]
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Calculate the area of the circle: The radius of the circle is 6 inches, so the area is given by the formula for the area of a circle: \[ \text{Area of circle} = \pi r^2 = \pi (6^2) = \pi \times 36 \approx 113.097 , \text{square inches} \]
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Calculate the area of the square: The side length of the square is 10 inches, so the area is: \[ \text{Area of square} = \text{side}^2 = 10^2 = 100 , \text{square inches} \]
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Find the total area occupied by the circle and the square: The circle and square are beside each other, so we can calculate their combined area: \[ \text{Total area (circle + square)} = \text{Area of circle} + \text{Area of square} \approx 113.097 + 100 = 213.097 , \text{square inches} \]
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Calculate the area that is not covered by the circle: The area of the rectangle that is not inside the circle is computed as follows: \[ \text{Area not in circle} = \text{Area of rectangle} - \text{Area of circle} \approx 720 - 113.097 \approx 606.903 , \text{square inches} \]
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Calculate the probability that a point is not inside the circle: The probability that a randomly chosen point in the rectangle is not inside the circle is: \[ P(\text{not in circle}) = \frac{\text{Area not in circle}}{\text{Area of rectangle}} \approx \frac{606.903}{720} \approx 0.842 \]
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Round the answer: Rounding 0.842 to the nearest hundredth, we get \(0.84\).
Thus, the final probability that a randomly placed point in the rectangle is not inside the circle is:
\[ \boxed{0.84} \]