A motor boat whose speed is $18\text{ km/h}$ in still water takes $1$ hour more to go $24\text{ km}$ upstream than to return downstream to the same spot. Find the speed of the stream.
Step-by-Step Solution
Key Concept: Let speed of stream be $x\text{ km/h}$. Upstream speed $= 18 - x$, downstream speed $= 18 + x$. Time equation: $\dfrac{24}{18-x} - \dfrac{24}{18+x} = 1$.
Stepwise Solution:
Let speed of stream $= x\text{ km/h}$. Upstream speed $= 18 - x$, downstream speed $= 18 + x$. [1.0 Mark]
Time upstream $t_1 = \dfrac{24}{18-x}$, Time downstream $t_2 = \dfrac{24}{18+x}$. Given $t_1 - t_2 = 1$. [1.0 Mark]
$\dfrac{24}{18-x} - \dfrac{24}{18+x} = 1 \Rightarrow 24 \left[ \dfrac{(18+x) - (18-x)}{(18-x)(18+x)} \right] = 1 \Rightarrow 24 \left[ \dfrac{2x}{324 - x^2} \right] = 1$. [1.5 Marks]
$48x = 324 - x^2 \Rightarrow x^2 + 48x - 324 = 0 \Rightarrow (x + 54)(x - 6) = 0$. [1.0 Mark]
Since speed of stream cannot be negative, $x = 6\text{ km/h}$ (rejecting $x = -54$). Speed of stream $= 6\text{ km/h}$. [0.5 Mark]
Marking Scheme:
• Setting up upstream/downstream speeds and times: 1.0 Mark
• Forming time difference equation: 1.0 Mark
• Simplifying to quadratic equation $x^2 + 48x - 324 = 0$: 1.5 Marks
• Factorising and finding $x = 6$: 1.0 Mark
• Concluding speed of stream is 6 km/h: 0.5 Mark
Correct Answer: