To find the probability that the car keys are not on the playground, we first need to determine the area of the park and the area of the playground.
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Calculate the area of the park: The park is rectangular with dimensions 80 ft × 90 ft. Therefore, the area of the park \( A_{\text{park}} \) is: \[ A_{\text{park}} = 80 \text{ ft} \times 90 \text{ ft} = 7200 \text{ ft}^2 \]
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Calculate the area of the playground: The playground is circular with a diameter of 50 ft. The radius \( r \) of the playground is half the diameter: \[ r = \frac{50 \text{ ft}}{2} = 25 \text{ ft} \] The area of the playground \( A_{\text{playground}} \) is given by the formula for the area of a circle: \[ A_{\text{playground}} = \pi r^2 = \pi (25 \text{ ft})^2 = 625\pi \text{ ft}^2 \]
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Calculate the area of the park not covered by the playground: The area of the region outside the playground within the park is: \[ A_{\text{not playground}} = A_{\text{park}} - A_{\text{playground}} = 7200 \text{ ft}^2 - 625\pi \text{ ft}^2 \]
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Calculate the probability that the keys are not on the playground: The probability \( P \) that the keys are not on the playground is the ratio of the area outside the playground to the area of the park: \[ P(\text{not on playground}) = \frac{A_{\text{not playground}}}{A_{\text{park}}} = \frac{7200 \text{ ft}^2 - 625\pi \text{ ft}^2}{7200 \text{ ft}^2} \]
To perform the calculation, we can approximate \( \pi \) as 3.14: \[ 625\pi \approx 625 \times 3.14 \approx 1963.75 \text{ ft}^2 \] So, \[ A_{\text{not playground}} \approx 7200 \text{ ft}^2 - 1963.75 \text{ ft}^2 \approx 5236.25 \text{ ft}^2 \] And now, \[ P(\text{not on playground}) \approx \frac{5236.25 \text{ ft}^2}{7200 \text{ ft}^2} \approx 0.7267 \] This gives us approximately \( 0.73 \).
Thus, the probability that the keys are not on the playground is approximately \( \boxed{0.73} \).