The boat had an average speed of ____ mph relative to the water?

Question

A boat traveled downstream a distance of 57 mi and then came right back. If the speed of the of the current was 12 mph and the total trip took 6 hours and 20 minutes find the average speed of the boat relative to the water?

Answer 1

√Distance = Speed X Time  -->  Time = Distance / Speed

For this problem you are given the total time is 6 hrs and 20 min  = 19/3 hr

The total time is equal to the time going down the river plus the time going up the river.  The following equation can be written:

Let v be the speed of the boat

57mi / (v+12)mph + 57mi / (v-12)mph = 19/3 hr

multiply each side of the equation by 1/57:  1 / (v+12) + 1 / (v-12) = 1/9

multiply each side of the equation by (v+12)(v-12):    (v-12) + (v+12) = 1/9(v²-144)  -->  2v = 1/9(v²) - 16

Collect terms on one side of the equal sign:  1/9(v²) - 2v - 16 = 0

Multiply both sides of the equation by 9:  v² - 18v - 144 = 0

This equation factors to (v-24)(v+6) = 0  -->  v = 24 or -6

The boat speed should not be negative, so choose v = 24 mph

 

Check:  57 / 36 + 57 / 12 = 19/3?  -->  19 / 12 + 57 / 12 = 76 / 12 = 19/3 = 19/3  check!