a motor boat whose speed is 18 km/h in still water takes 1 hour more to go 24 km upstream than to return downstream to the same spot. find the speed of stream.
Given Information:
A motorboat travels at 18 km/h in still water. It takes 1 hour more to travel 24 km upstream than to return downstream to the same spot.
Let’s denote the speed of the stream as ( x ) km/h.
Calculating the time taken to go upstream:
When the boat is going upstream, the effective speed is ( 18 - x ) km/h (18 km/h in still water minus the speed of the stream).
Time taken to travel 24 km upstream:
[ \frac{24}{18 - x} ]
Calculating the time taken to go downstream:
When the boat is going downstream, the effective speed is ( 18 + x ) km/h (18 km/h in still water plus the speed of the stream).
Time taken to travel 24 km downstream:
[ \frac{24}{18 + x} ]
Given that it takes 1 hour more to go upstream than to return downstream:
[ \frac{24}{18 - x} = \frac{24}{18 + x} + 1 ]
Solving this equation will give us the speed of the stream. Let’s solve it step by step:
[ \frac{24}{18 - x} = \frac{24}{18 + x} + 1 ]
[ \frac{24}{18 - x} = \frac{24 + (18 + x)}{18 + x} ]
[ \frac{24}{18 - x} = \frac{42 + x}{18 + x} ]
[ 24(18 + x) = (42 + x)(18 - x) ]
[ 432 + 24x = 756 - 18x + 42x - x^2 ]
[ x^2 - 60x + 324 = 0 ]
[ (x - 6)(x - 54) = 0 ]
This equation has two solutions: ( x = 6 ) and ( x = 54 ). However, the speed of the stream cannot be negative, so the speed of the stream is ( x = 6 ) km/h.
Therefore, the speed of the stream is ( 6 ) km/h.