To solve the problem, we'll start with Part 1.
Part 1: Time until the cheetah catches the gazelle
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Convert speeds from km/h to m/s: \[ \text{Speed of cheetah} = 101 \text{ km/h} = \frac{101 \times 1000}{3600} \approx 28.06 \text{ m/s} \] \[ \text{Speed of gazelle} = 78.4 \text{ km/h} = \frac{78.4 \times 1000}{3600} \approx 21.78 \text{ m/s} \]
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Determine the relative speed of the cheetah compared to the gazelle: \[ \text{Relative speed} = \text{Speed of cheetah} - \text{Speed of gazelle} = 28.06 \text{ m/s} - 21.78 \text{ m/s} = 6.28 \text{ m/s} \]
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Calculate the time it takes for the cheetah to close the distance: The cheetah is 82.7 m behind the gazelle, so we can use the formula: \[ \text{Time} = \frac{\text{Distance}}{\text{Relative speed}} = \frac{82.7 \text{ m}}{6.28 \text{ m/s}} \approx 13.15 \text{ s} \]
Thus, the cheetah will catch the gazelle in approximately 13.15 seconds.
Part 2: Minimum distance for the gazelle to escape
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Calculate the distance the cheetah can run in 7.5 seconds: \[ \text{Distance cheetah can run} = \text{Speed of cheetah} \times \text{Time} = 28.06 \text{ m/s} \times 7.5 \text{ s} \approx 210.45 \text{ m} \]
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Calculate the distance the gazelle can run in the same 7.5 seconds: \[ \text{Distance gazelle can run} = \text{Speed of gazelle} \times \text{Time} = 21.78 \text{ m/s} \times 7.5 \text{ s} \approx 163.35 \text{ m} \]
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Find the minimum distance the gazelle must be ahead: The gazelle needs to have a minimum distance that ensures it is ahead after both have run for 7.5 seconds. The minimum distance can be calculated as the difference in distances they cover in 7.5 seconds: \[ \text{Minimum lead} = \text{Distance cheetah can run} - \text{Distance gazelle can run} = 210.45 \text{ m} - 163.35 \text{ m} \approx 47.1 \text{ m} \]
Thus, to ensure a chance of escape, the gazelle must be at least 47.1 meters ahead of the cheetah.
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