Question

A boat whose speed is 15 km/hour in still water goes 36 km with stream and 24 km against the stream in a total of 4 hours. Find the speed of the stream.

Answer 1

Answer:

$v_s=0\ km/h$

$v_s=3\ km/h$

Explanation:

Relative Speed

A boat traveling in still waters has a speed v_b. If now an opposite stream appears, then the apparent speed of the boat will be less than before because the relative speed is the subtraction of both speeds. Thus we say

$v_{t1}=v_b-v_s$

Conversely, if the stream goes with the boat, both speeds are added.

$v_{t2}=v_b+v_s$

where $v_t$ it the total or real speed of the boat respect to the ground, $v_b$ is the speed of the boat in still water, and $v_s$ is the speed of the stream.

The boat goes 36 km with stream. The time it took for doing so is:

$\displaystyle t_1=\frac{36}{v_b+v_s}$

The boat goes 24 km against the stream in

$\displaystyle t_2=\frac{24}{v_b-v_s}$

The problem states that

$t_1+t_2=4\ hours$

$\displaystyle \frac{36}{v_b+v_s}+\frac{24}{v_b-v_s}=4$

Knowing $v_b=15$

$\displaystyle \frac{36}{15+v_s}+\frac{24}{15-v_s}=4$

Operating

$\displaystyle 36(15-v_s)+24(15+v_s)=4(15+v_s)(15-v_s)$

Rearranging and simplifying

$225-3v_s=225-v_s^2$

Factoring

$v_s(v_s-3)=0)$

We get  

$v_s=0,\ v_s=3$

Both solutions are possible, i.e.

1. The stream does not exist and the boat travels the total distance of 60 Km in 4 hours

2. The stream has a speed of 3 Km/h and the boat travels 36 km in 2 hours and 24 km in 2 hours