To determine the distance after which the fore wheel makes 100 more revolutions than the hind wheel, we will first find the distance traveled by each wheel in 100 revolutions.
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Circumference of the wheels:
- Fore wheel: \( C_f = 10 \) feet
- Hind wheel: \( C_h = 12 \) feet
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Distance traveled in 100 revolutions:
- Distance covered by the fore wheel in 100 revolutions: \[ D_f = 100 \times C_f = 100 \times 10 = 1000 \text{ feet} \]
- Distance covered by the hind wheel in \( x \) revolutions: Let \( x \) be the number of revolutions the hind wheel makes. The distance covered by the hind wheel in \( x \) revolutions is: \[ D_h = x \times C_h = x \times 12 \]
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Equating the distances: We want to find the distance \( D \) such that the fore wheel has made 100 more revolutions than the hind wheel: \[ \frac{D}{C_f} = \frac{D}{10} = \text{Number of revolutions of the fore wheel} \] \[ \frac{D}{C_h} = \frac{D}{12} = \text{Number of revolutions of the hind wheel} \]
According to the problem statement, we have: \[ \frac{D}{10} = \frac{D}{12} + 100 \]
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Clearing the fractions: Multiply through by 60 (the least common multiple of 10 and 12): \[ 6D = 5D + 6000 \] Simplifying this gives: \[ 6D - 5D = 6000 \] \[ D = 6000 \text{ feet} \]
Thus, the distance after which the fore wheel will make 100 revolutions more than the hind wheel is 6000 feet.