Question

Swati can row a boat on still water at a speed of 5 km/hr. However, on a given river, it takes her 1 hour more to row the boat 12 km upstream than downstream. One day, Swati rows the boat on this river from X to Y, which is N km upstream from X. Then she rows back to X immediately. If she takes at least 2 hours to complete this round trip, what is the minimum possible value of N?

• A. 3.9 km
• B. 2.2 km
• C. 3.8 km
• D. 3.6 km
• E. 4.8 km

Hint:

In geometry problems with diagrams, identify the key relationships (similar triangles, right angles, parallel lines) to set up the equations.

Answer 1

Answer: (E)

Step 1: Understanding the Concept:
Let boat speed $v = 5$ km/hr and stream speed be $s$.
Upstream speed $= 5 - s$, Downstream speed $= 5 + s$.
Step 2: Key Formula or Approach:
1. $\frac{12}{5-s} - \frac{12}{5+s} = 1$ (to find $s$).
2. Round trip time $= \frac{N}{5-s} + \frac{N}{5+s} \geq 2$.
Step 3: Detailed Explanation:
Solve for $s$:
$12(\frac{5+s - (5-s)}{25-s^2}) = 1$.
$12(\frac{2s}{25-s^2}) = 1 \implies 24s = 25 - s^2 \implies s^2 + 24s - 25 = 0$.
Roots are $s = 1$ and $s = -25$. Since $s>0$, $s = 1$ km/hr.
Now solve for $N$:
Upstream speed $= 4$, Downstream speed $= 6$.
$\frac{N}{4} + \frac{N}{6} \geq 2$.
$\frac{3N + 2N}{12} \geq 2 \implies \frac{5N}{12} \geq 2$.
$5N \geq 24 \implies N \geq 4.8$.
Step 4: Final Answer:
The minimum value of N is 4.8 km.