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.
When the boat goes upstream, the current slows it down. So, upstream speed = $x - 1$ km/h.
When the boat returns downstream, the current aids its speed. So, downstream speed = $x + 1$ km/h.
Distance each way = 3.5 km.
Time taken upstream = $\dfrac{3.5}{x - 1}$
Time taken downstream = $\dfrac{3.5}{x + 1}$
Total time = $1$ hour $12$ minutes = $\dfrac{72}{60} = \dfrac{6}{5}$ hours
So, \[ \frac{3.5}{x - 1} + \frac{3.5}{x + 1} = \frac{6}{5} \]
Multiply both sides by 5 to eliminate the denominator:
\[ 17.5\left(\frac{1}{x - 1} + \frac{1}{x + 1}\right) = 6 \]
Now solve the expression inside the parentheses:
\[ \frac{1}{x - 1} + \frac{1}{x + 1} = \frac{(x + 1) + (x - 1)}{x^2 - 1} = \frac{2x}{x^2 - 1} \]
Substitute back: \[ 17.5.\frac{2x}{x^2 - 1} = 6 \Rightarrow \frac{35x}{x^2 - 1} = 6 \]
Cross-multiply: \[ 35x = 6x^2 - 6 \Rightarrow 6x^2 - 35x - 6 = 0 \]
Now solve the quadratic equation using the quadratic formula:
\[ x = \frac{35 \pm \sqrt{(-35)^2 - 4.6.(-6)}}{2.6} = \frac{35 \pm \sqrt{1225 + 144}}{12} = \frac{35 \pm \sqrt{1369}}{12} \]
Since $\sqrt{1369} = 37$,
\[ x = \frac{35 + 37}{12} = \frac{72}{12} = 6 \text{or} x = \frac{35 - 37}{12} = -\frac{2}{12} = -\frac{1}{6} \]
Speed cannot be negative, so discard the negative root.
Final answer: speed of the boat in still water is 6 km/h.