word problem!?

Question

A Cruise liner travels with the water current for 1000 miles. On the way to the destination, the water current helps and makes the travel time shorter. The speed of the water current is 5 miles an hour. The round trip takes 45 hours. How fast would the speed of the Cruise liner be if there was no current?

Answer 1

velocity * time = distance

Let V = the velocity of the boat in still water. T1 is the time to the destination and T2 the time back.

( V + 5 ) * T1 = 1000
( V - 5 ) * T2 = 1000
T1 + T2 = 45

To solve this set of three equations with three unknowns, we need to make some substitutions. Use the first two equations to get expressions for T1 and T2...

( V + 5 ) * T1 = 1000 so T1 = 1000 / ( V + 5 )
( V - 5 ) * T2 = 1000 so T2 = 1000 / ( V - 5 )

...and substitute those into the third equation.

T1 + T2 = 45
1000 / ( V + 5 ) + 1000 / ( V - 5 ) = 45

Multiply through by ( V + 5 ) ( V - 5 )...

( 1000 ) ( V - 5 ) + ( 1000 ) ( V + 5 ) = ( 45 ) ( V + 5 ) ( V - 5 )

and simplify...

1000 V - 5000 + 1000 V + 5000 = 45 ( V² - 25 )
2000 V = 45 V² - 1125
45 V² - 2000 V - 1125 = 0
9 V² - 400 V - 225 = 0

This is a quadratic equation so

V = ( 400 +/- sqrt ( 400² - 4 ( 9 ) ( -225 ) ) / ( 2 ) ( 9 )

The positive solution is V = 45, so that's the speed of the boat in still water. The speed was 50 mph on the way out and only 40 mph on the way back.

Checking:

1000 miles @ 50 mph = 20 hours
1000 miles @ 40 mph = 25 hours
total time is 45 hours as expected.