Question

A boat goes 3.5 km upstream and then returns. Total time taken is 1 hour and 12 minutes. If the speed of the current is 1 km/h, then find the speed of the boat in still water.

Hint:

When solving for time with current, adjust speeds for upstream/downstream. Always convert minutes to hours for consistency in time units.

Answer 1

Let the speed of the boat in still water be $x$ km/h.
Upstream speed is $x - 1$ km/h. Downstream speed is $x + 1$ km/h.
The distance for each journey is 3.5 km.
Time upstream = $\dfrac{3.5}{x - 1}$ hours.
Time downstream = $\dfrac{3.5}{x + 1}$ hours.
Total time is 1 hour 12 minutes, which is $\dfrac{72}{60} = \dfrac{6}{5}$ hours.
This leads to the equation:
\[\frac{3.5}{x - 1} + \frac{3.5}{x + 1} = \frac{6}{5}\]
Multiplying by 5 yields:
\[17.5\left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) = 6\]
Simplifying the expression in the parentheses:
\[\frac{1}{x - 1} + \frac{1}{x + 1} = \frac{2x}{x^2 - 1}\]
Substituting back gives:
\[17.5.\frac{2x}{x^2 - 1} = 6 \Rightarrow \frac{35x}{x^2 - 1} = 6\]
Cross-multiplication results in:
\[35x = 6x^2 - 6 \Rightarrow 6x^2 - 35x - 6 = 0\]
Using the quadratic formula, $x = \dfrac{35 \pm \sqrt{(-35)^2 - 4(6)(-6)}}{2(6)}$:
\[x = \frac{35 \pm \sqrt{1225 + 144}}{12} = \frac{35 \pm \sqrt{1369}}{12}\]
Since $\sqrt{1369} = 37$, the solutions are:
\[x = \frac{35 + 37}{12} = 6 \text{ or } x = \frac{35 - 37}{12} = -\frac{1}{6}\]
The speed cannot be negative, so $x = -\frac{1}{6}$ is discarded.
Therefore, the speed of the boat in still water is 6 km/h.