To find the probability that a dart lands inside the square but not on the circular dartboard, we need to follow these steps:
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Calculate the area of the circle: Given in the problem, the area of the circle is unspecified, but we are asked to assume it corresponds to the value calculated using \( \pi \) and the radius. However, we will not use this since the area of the circle is not explicitly given, but we can still check the calculations as we compute both areas.
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Calculate the area of the square: The area of the square is given as 64 square inches.
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Determine the relationship between the circle and the square: The circle is inscribed in the square, meaning that the diameter of the circle equals the side length of the square.
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The formula for the area of a square with side length \( s \) is: \[ \text{Area}_{\text{square}} = s^2 \] Thus, if the area is 64 square inches: \[ s^2 = 64 \implies s = 8 , \text{inches} \]
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Consequently, the diameter of the circle also equals the square's side, which is 8 inches.
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The radius \( r \) of the circle is: \[ r = \frac{8}{2} = 4 , \text{inches} \]
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Calculate the area of the circle: \[ \text{Area}_{\text{circle}} = \pi r^2 = 3.14 \times (4^2) = 3.14 \times 16 = 50.24 , \text{square inches} \]
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Calculate the area of the region outside the circle but inside the square: \[ \text{Area outside circle} = \text{Area}{\text{square}} - \text{Area}{\text{circle}} = 64 - 50.24 = 13.76 , \text{square inches} \]
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Calculate the probability that the dart lands in the area outside the circle but inside the square: \[ \text{Probability} = \frac{\text{Area outside circle}}{\text{Area of square}} = \frac{13.76}{64} \]
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Calculate this value: \[ \frac{13.76}{64} \approx 0.215 \]
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Convert this to a percentage: \[ 0.215 \times 100 \approx 21.5% \]
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Round to the nearest percent: \[ \text{Probability} \approx 22% \]
Thus, the probability that the dart lands inside the square but not on the circular dartboard is 22%.