To determine the sample space of the area where the souvenir was lost, we need to calculate the total area of the rectangular garden and then subtract the areas of the circular pond and the semicircular picnic area.
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Area of the rectangular garden: \[ \text{Area}_{\text{rectangle}} = \text{length} \times \text{width} = 80, \text{ft} \times 90, \text{ft} = 7200, \text{ft}^2 \]
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Area of the circular pond: \[ \text{Area}_{\text{circle}} = \pi r^2 = \pi (25, \text{ft})^2 = \pi (625, \text{ft}^2) \approx 1963.5, \text{ft}^2 \]
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Area of the semicircular picnic area: \[ \text{Area}{\text{semicircle}} = \frac{1}{2} \pi r^2 \] Since the diameter is 50 ft, the radius \( r \) is: \[ r = \frac{50, \text{ft}}{2} = 25, \text{ft} \] Therefore, the area is: \[ \text{Area}{\text{semicircle}} = \frac{1}{2} \pi (25, \text{ft})^2 = \frac{1}{2} \pi (625, \text{ft}^2) \approx \frac{1}{2} (1963.5, \text{ft}^2) \approx 981.75, \text{ft}^2 \]
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Total area of the garden where the souvenir could be lost: \[ \text{Area}{\text{sample space}} = \text{Area}{\text{rectangle}} - \text{Area}{\text{circle}} - \text{Area}{\text{semicircle}} \] Calculating this: \[ \text{Area}_{\text{sample space}} = 7200, \text{ft}^2 - 1963.5, \text{ft}^2 - 981.75, \text{ft}^2 \] \[ = 7200 - 1963.5 - 981.75 \approx 7200 - 2945.25 = 4254.75, \text{ft}^2 \]
So the area that represents where the souvenir could be lost is approximately: \[ \text{Area}_{\text{sample space}} \approx 4254.76, \text{ft}^2 \]
Therefore, the answer is \( \boxed{4,254.76, \text{ft}^2} \).