Question

A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Another boat takes a total of 6 hours to travel from port B to port A and return to port B . If the speeds of the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port A to port B is

• A. \(3(\sqrt 5-1)\)
• B. \(3(3+\sqrt 5)\)
• C. \(3(3-\sqrt5)\)
• D. \(12(\sqrt5-2)\)

Answer 1

The Correct Option is C

Let the speed of the first boat be $b$, the second boat be $s$, and the river's speed be $r$.
Let the distance between points A and B be $d$.

From the question: 
$\Rightarrow d = 2(b + r)$ and $d = 3(b - r)$

Solving both equations:
$\Rightarrow b + r = \frac{d}{2}$ and $b - r = \frac{d}{3}$

Subtracting the two equations:
$\Rightarrow (b + r) - (b - r) = \frac{d}{2} - \frac{d}{3}$
$\Rightarrow 2r = \frac{3d - 2d}{6} = \frac{d}{6} \Rightarrow r = \frac{d}{12}$

Now, using the time relation for the second boat:
$\frac{d}{s + r} + \frac{d}{s - r} = 6$

Substitute $r = \frac{d}{12}$:
$\Rightarrow \frac{d}{s + \frac{d}{12}} + \frac{d}{s - \frac{d}{12}} = 6$

Multiply numerator and denominator by 12 to simplify:
Let’s multiply entire equation by the LCM to simplify:

Multiply numerator and denominator appropriately:
$\Rightarrow \frac{d(12)}{12s + d} + \frac{d(12)}{12s - d} = 6$

Multiply both sides by $(12s + d)(12s - d)$: 
$\Rightarrow 12d(12s - d) + 12d(12s + d) = 6(144s^2 - d^2)$

Simplify:
$144ds - 12d^2 + 144ds + 12d^2 = 6(144s^2 - d^2)$
$\Rightarrow 288ds = 864s^2 - 6d^2$

Bring all terms to one side:
$\Rightarrow 144s^2 - 48ds - d^2 = 0$

This is a quadratic in $s$, solve using quadratic formula:
$s = \frac{48d + \sqrt{(48d)^2 + 4(144)(d^2)}}{2 \cdot 144}$
$= d\left(\frac{48 + \sqrt{48^2 + 4 \cdot 144}}{2 \cdot 144}\right)$

Simplifying:
$s = d\left(\frac{1}{6} + \frac{\sqrt{5}}{12}\right)$

Now, compute the required value:
$\frac{d}{s + r} = \frac{d}{\frac{d}{6} + \frac{d\sqrt{5}}{12} + \frac{d}{12}}$

$\Rightarrow \frac{1}{\frac{1}{6} + \frac{\sqrt{5}}{12} + \frac{1}{12}} = \frac{1}{\frac{3 + \sqrt{5}}{12}} = \frac{12}{3 + \sqrt{5}}$

Rationalize the denominator:
$\Rightarrow \frac{12}{3 + \sqrt{5}} \cdot \frac{3 - \sqrt{5}}{3 - \sqrt{5}} = \frac{12(3 - \sqrt{5})}{9 - 5} = \frac{12(3 - \sqrt{5})}{4}$

$\Rightarrow 3(3 - \sqrt{5})$

Therefore, the correct option is (C): $\boxed{3(3 - \sqrt{5})}$