To determine how long it takes for the cheetah to catch the gazelle, we can start by calculating their speeds in meters per second (m/s):
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Convert the speeds from km/h to m/s: \[ \text{Cheetah's speed} = 101 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{101 \times 1000}{3600} \approx 28.06 \text{ m/s} \]
\[ \text{Gazelle's speed} = 78.4 \text{ km/h} \times \frac{1000 \text{ m}}{1 \text{ km}} \times \frac{1 \text{ h}}{3600 \text{ s}} = \frac{78.4 \times 1000}{3600} \approx 21.78 \text{ m/s} \]
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Determine the relative speed: \[ \text{Relative speed} = \text{Cheetah's speed} - \text{Gazelle's speed} = 28.06 \text{ m/s} - 21.78 \text{ m/s} = 6.28 \text{ m/s} \]
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Determine the initial distance between them: The gazelle is 82.7 m ahead of the cheetah.
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Calculate the time it takes for the cheetah to catch the gazelle: Using the formula time = distance / speed: \[ \text{time} = \frac{\text{distance}}{\text{relative speed}} = \frac{82.7 \text{ m}}{6.28 \text{ m/s}} \approx 13.15 \text{ s} \]
Thus, the time it takes for the cheetah to catch the gazelle is approximately \(\boxed{13.15}\) seconds.