Question

A factory $D$ is to be connected by a road with a straight railway line on which a town $A$ is situated. The distance $DB$ of the factory to the railway line is $5\sqrt{3}$ km. Length $AB$ of the railway line is $20km$. Freight charges on the road are twice the charges on the railway. At what point $P(AP<AB)$ on the railway line should the road $DP$ be connected so as to ensure minimum freight charges from the factory to the town?

• A. $BP=5km$
• B. $AP=5km$
• C. $BP=7.5km$
• D. None of these

Answer 1

Answer: (B)

$AP=5km$

Considering the given figure
Let R rupees per km be the freight for rail, then that by road it will be 2R per km. If C be the total cost then
$C=R[2\sqrt{x^2+75}+20-x]$
$\frac{dC}{dx}=R.[\frac{4x}{2\sqrt{x^2+75}}-1]$
$=0$
Or
$2x=\sqrt{x^2+75}$
Or
$4x^2=x^2+75$
$3x^2=75$
$x^2=25$
$x=5$ and $x=-5$ (not possible).
Hence Cost is minimum at $x=5$.