To solve the problem with the information given about the distance traveled by the two runners, let's define the scenarios you're referring to.
Let’s denote:
- Runner 1's distance after 6 seconds as \(d_1\).
- Runner 2's distance after \(t_2\) seconds as \(d_2\).
- We know that Runner 2 has traveled 10 feet after \(t_2\) seconds.
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We can find the speed of Runner 2. If they travel 10 feet in \(t_2\) seconds, then their speed is: \[ \text{Speed of Runner 2} = \frac{10 \text{ feet}}{t_2 \text{ seconds}}. \]
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For Runner 1, we need to know how far they traveled after 6 seconds. Assume \(d_1\) is the distance. To compare the two runners, we need to find the time it takes each to complete 100 feet.
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To find the time for Runner 1 to run 100 feet: \[ t_1 = \frac{100 \text{ feet}}{\text{Speed of Runner 1}}. \] If Runner 1 has traveled \(d_1\) feet in 6 seconds, the speed of Runner 1 would be: \[ \text{Speed of Runner 1} = \frac{d_1}{6}. \]
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To finish the equation for Runner 1: \[ t_1 = \frac{100}{\frac{d_1}{6}} = \frac{600}{d_1} \text{ seconds}. \]
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For Runner 2:
- First, we need to assume or know how many seconds it took them to reach 10 feet. We can set a similar equation: \[ t_2 = \frac{100}{\frac{10}{t_2}} = \frac{100 \times t_2}{10} = 10 \cdot t_2 \text{ seconds}. \] However, since \(t_2 = \frac{10}{\text{Speed of Runner 2}}\), we didn't need to derive it like this if we substitute. Instead, we will just note that Runner 2 runs a distance of 10 feet in \( t_2 \) seconds.
Now, to determine which runner would finish a race of 100 feet first, you'll compare \(t_1\) and \(t_2\).
To conclude, based on the distances each runner can cover given their speed determined by their distance during the sampled time frames, you should fill in the gaps using provided or known distances:
- For Runner 1 after 6 seconds, insert \(d_1\).
- For Runner 2, count the seconds it takes them based on the desired speed that can be inferred from 10 feet.
Without the specific distance for Runner 1 after 6 seconds and the time for Runner 2 to travel 10 feet, you would substitute those variables or numbers, determined, to decide which runner would win the race to 100 feet.
Please fill out any missing details, like:
- Distance for Runner 1 after 6 seconds,
- Time taken for Runner 2 to reach 10 feet, to arrive at a complete answer.
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