Understanding Boat and Stream Speed Problems

This problem involves calculating the speed of a stream given the distance traveled upstream and downstream, the total time taken, and the speed of the boat in still water. We need to use the concepts of relative speed for upstream and downstream journeys.

Defining Variables

• Let $b$ represent the speed of the boat in still water.
• Let $s$ represent the speed of the stream.

Formulating Speeds and Times

When the boat travels upstream (against the current), its effective speed is reduced. The speed upstream is calculated as:

Speed Upstream = $b - s$

When the boat travels downstream (with the current), its effective speed is increased. The speed downstream is calculated as:

Speed Downstream = $b + s$

The relationship between distance, speed, and time is given by: Time = Distance / Speed.

Setting Up the Equation

We are given:

• Distance upstream = 35 km
• Distance downstream = 35 km
• Total time taken = 8 hours
• Speed of the boat in still water, $b$ = 9 km/h

The time taken for the upstream journey is:

Time Upstream = $\frac{\text{Distance Upstream}}{\text{Speed Upstream}} = \frac{35}{b - s}$

The time taken for the downstream journey is:

Time Downstream = $\frac{\text{Distance Downstream}}{\text{Speed Downstream}} = \frac{35}{b + s}$

The total time is the sum of the time taken for both journeys:

Total Time = Time Upstream + Time Downstream

Substituting the given values:

$\frac{35}{b - s} + \frac{35}{b + s} = 8$

Solving for Stream Speed

Now, substitute the value of $b = 9$ km/h into the equation:

$\frac{35}{9 - s} + \frac{35}{9 + s} = 8$

To solve this equation, we first find a common denominator for the fractions on the left side, which is $(9 - s)(9 + s)$.

$\frac{35(9 + s) + 35(9 - s)}{(9 - s)(9 + s)} = 8$

Simplify the numerator:

$35 \times 9 + 35s + 35 \times 9 - 35s = 315 + 35s + 315 - 35s = 630$

Simplify the denominator using the difference of squares formula ($(a-b)(a+b) = a^2 - b^2)$:

$(9 - s)(9 + s) = 9^2 - s^2 = 81 - s^2$

Substitute the simplified numerator and denominator back into the equation:

$\frac{630}{81 - s^2} = 8$

Now, cross-multiply:

$630 = 8(81 - s^2)$

$630 = 648 - 8s^2$

Rearrange the equation to solve for $s^2$:

$8s^2 = 648 - 630$

$8s^2 = 18$

$s^2 = \frac{18}{8}$

$s^2 = \frac{9}{4}$

Take the square root of both sides to find the speed of the stream, $s$. Since speed must be positive:

$s = \sqrt{\frac{9}{4}}$

$s = \frac{3}{2}$

$s = 1.5$ km/h

Conclusion

The speed of the stream is 1.5 km/h.