To calculate the geometric probability of hitting the circle on the corn hole board, we first need to find the area of both the board and the circular hole.
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Calculate the area of the board: The dimensions of the board are given as 48 inches by 24 inches. \[ \text{Area of the board} = \text{length} \times \text{width} = 48 , \text{inches} \times 24 , \text{inches} = 1152 , \text{square inches} \]
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Calculate the area of the circle: The diameter of the circle is given as 6 inches, so the radius \( r \) is: \[ r = \frac{\text{diameter}}{2} = \frac{6 , \text{inches}}{2} = 3 , \text{inches} \] The area of the circle is calculated using the formula for the area of a circle, \( A = \pi r^2 \): \[ \text{Area of the circle} = \pi (3 , \text{inches})^2 = \pi \times 9 , \text{inches}^2 = 9\pi , \text{square inches} \]
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Calculate the geometric probability: The geometric probability \( P \) of hitting the circle when tossing an object onto the board is the area of the circle divided by the area of the board: \[ P = \frac{\text{Area of the circle}}{\text{Area of the board}} = \frac{9\pi , \text{square inches}}{1152 , \text{square inches}} \]
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Calculating the numerical value: We can use \( \pi \approx 3.14 \) for a numerical approximation, or you can keep it in terms of \( \pi \) if required. \[ P = \frac{9 \times 3.14}{1152} \approx \frac{28.26}{1152} \approx 0.0245 \] So, \[ P \approx 0.0245 \]
Alternatively, if kept in terms of \(\pi\): \[ P = \frac{9\pi}{1152} \approx \frac{0.0283}{1} \text{ (keep \(\pi\) in the numerator for exact probabilities)} \]
Thus, the geometric probability that the object will hit the circular hole on the board is approximately 0.0245 or \( \frac{9\pi}{1152} \) in exact terms.