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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 $5 \sqrt{3} k m$ . Length AB of the railway line is 20km. Freight charges on the road are twice the charges on the railway. The 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 is

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a

BP = 5km

b

AP = 5km

c

BP = 7.5km

d

AP = 15 km

answer is A, D.

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Detailed Solution

Let then charges for railways line be k Rs /km
Now total freight charges, $T = k x + 2 x \sqrt{{(20 - x)}^{2} + 75}$
Let $\frac{d T}{d x} = 0$
or $k + 2 k \frac{2 (x - 20)}{2 \sqrt{{(x - 20)}^{2} + 75}} = 0$
or $4 {(x - 20)}^{2} = 75 + {(x - 20)}^{2}$
or ${(x - 20)}^{2} = 25$
or $x = 25, 15, o r x = 15 (a s A P < A B)$
or $P B = A B - A P = 20 - 15 = 5 k m$

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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 53km . Length AB of the railway line is 20km. Freight charges on the road are twice the charges on the railway. The 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 is