Power cable question

A power cable is to be run from a power plant on the bank of a river 900 meters wide to a factory that is located 3000 meters downstream on the opposite bank. If the cost of laying cable under water is Rs. 5 per meter and that of laying overhead on land is Rs. 4 per meter, find the point downstream where the cable is to cut across the river.

This question is related to TCS Interview

abhijit_de2000
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Jul 16th, 2008
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Raze2dust
Profile Answers by Raze2dust

Nov 9th, 2008

Ok..I realize this needs a bit of math(calculus, to be specific)..not sure if theres any other way out..

A<--------x------>C D
------------------------------------
^
|
900m
|
V
------------------------------------
B
<---------3000m---------------->

Let the path be A-C-B and angle BCD=t
Now CD=3000-x= 900/tan(t)=900 cot(t). So x=AB=3000-900cot(t)
BD=900
So, BC= 900/sin(t) = 900cosec(t)

Total cost = f = 4*AB+5*BC = 4(3000-900cot(t))+5(900cosec(t))
=12000-3600cot(t)+4500cosec(t)

We need to minimize f. So put df/dx=0
gives 3600cosec2(t)-4500cosec(t)cot(t)=0
3600cosec(t)=4500cot(t)..write cosec as 1/sin and cot as cos/sin..
gives cos(t)=4/5. So tan(t)=3/4.
BD/CD=3/4
900/(3000-x)=3/4
3600=9000-3x
So x=1800m

I'll try to think of a simpler method though...

agkjack
Profile Answers by agkjack

Nov 17th, 2009

I do not think it is complex. They give cost to plant the cable in ground and water both also
Let assume the length across the river is x;

By given cost
5x=4*3000+5*900
so x=3300

3300/2=1650

so the point is 1650m from x axis, and 450m(half of distance bet banks) from y axis
(450,1650)

smokincigar07
Profile Answers by smokincigar07

Feb 20th, 2010

<(3000-x)->C
A------------------------------------------------------
          ^       |
           |       |
          900 |
           |       |
           |       |
           |       |
--------------------------------------------------------
                    <-- x -->B

CB=(900^2+x^2)^1/2

cost of laying cable on land=4*(3000-x)
cost of laying cable under water=5*(900^2+x^2)^1/2

total cost T=4*(3000-x)+5*((900^2+x^2)^1/2

to minimize T
dT/dx=0

this will give us x=1200

therefore the distance from A to C (where the cable cut though the river) is
3000-1200=1800 metres