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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.


  
Total Answers and Comments: 2 Last Update: November 17, 2009     Asked by: abhijit_de2000 
  
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November 09, 2008 00:58:59   #1  
Raze2dust Member Since: November 2008   Contribution: 3    

RE: Power cable question
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...

 
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November 17, 2009 02:05:33   #2  
agkjack Member Since: November 2009   Contribution: 1    

RE: Power cable question
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)

 
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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. (a) 450 (b) 3900 (c) 1800 (d) 2100 (e) 2700
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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 
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