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Question
Utility poles are to be placed at equal intervals.  If 4 fewer poles were used per mile, the space between every 2 consecutive poles would be increased by 12 feet.

Calculate number of poles per mile.

Note:  For each mile, include the pole at the beginning but not the pole at the end.

Thanks.

Let n be the number of poles used per mile.
1 mile = 5280 feet
The distance between consecutive poles is (5280/n) feet.

"If 4 fewer poles were used per mile, the space between every 2 consecutive poles would be increased by 12 feet."
5280/(n-4) = 12 + 5280/n

divide both sides by 12
440/(n-4) = 1 + 440/n
440/(n-4) = (n+440)/n
440n = (n+440)(n-4)
440n = n² + 436n - 1760
n² - 4n - 1760 = 0

n = [4 ± √(4² - 4(1)(-1760))]/(2·1)
= [4 ± √7056]/2
= [4 ± 84]/2
= -40, 44 but n can't be negative, so n = 44.

There are 44 poles per mile.

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