# math

A boat traveling downstream a distance of 39 miles then came right back.If the speed of the current was 12 mph and the total trip took 4 hours 20 minutes,find the average speed of the boat relative to the water.

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1. speed of boat --- x mph

time to go downstream = 39/(x+12)
time to go upstream = 39/(x-12) , clearly x > 12

39/(x+12) + 39/(x-12) = 4 1/3 = 13/3
(39(x-12) + 39(x+12))/(x^ - 144) = 13/3
78x/(x^2 - 144) = 13/3
13x^2 - 1872 = 234x
13x^2 - 243x - 1872 = 0
x^2 - 18x - 144 = 0
completing the square, (faster than the formula for this one)
x^2 - 18x + 81 = 144 + 81
(x-9)^2 = 225
x-9 =±15
x = 9 ±15 = 24 or an inadmissable -6

the boat's speed was 24 mph

check:
39/36 + 39/12
= 156/36
= 13/3 or 4 1/3 or 4 hrs 20 minutes

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