Question

A small island is situated $5$ km from the nearest point $A$ on the straight shoreline of a large lake. If a swimmer on this island can swin at a rate of $2$ kmph and can walk $3$ kmph on the land then, how far away should he land from point $A$ in order to arrive at his house $11$ km down the shore from $A$ in the least time?

• A. $2$ km,
• B. $4\sqrt{5}$ km,
• C. $2\sqrt{5}$ km
• D. $4$ km.

Answer 1

The correct option is C $2\sqrt{5}$ km


Distance travelled in the sea
$d_s=\sqrt{5^2+x^2}$
Distance covered on land
$d_L=11-x$
time taken on sea $=t_s=\frac{\sqrt{25+x^2}}{2}$
Time taken on land $=t_L=\frac{11-x}{3}$
Total time taken $=t=\frac{\sqrt{25+x^2}}{2}+\frac{11-x}{3}$
On taking derivative to find minimum time, we have -
$\frac{dt}{dx}=\frac{1}{2}\times \frac{1}{2\sqrt{25+x^2}}\times 2x+\left(\frac{-1}{3}\right)=0\Rightarrow 3x-2\sqrt{25+x^2}=0$
On re-arranging & squaring, we have -
$9x^2=4(25+x^2)\Rightarrow 5x^2=4\times 25$
$\Rightarrow x=2\sqrt{5}$ km