# Problem involving calculus

1. Sep 20, 2010

### the_morbidus

1. The problem statement, all variables and given/known data
A new cottage is built across the river and 300 m downstream from the nearest telephone relay station. The river is 120m wide. In order to wire the cottage for phone service, wire will be laid across the river under water, and along the edge of the river above ground. The cost to lay wire under water is $15 per m and the cost to lay wire above ground is$10 per m. How much wire should be laid under water to minimize cost?

2. Relevant equations
Pythagoras theorem a^2 +b^2 = c^2

3. The attempt at a solution

ok so i think the wire going across the river will create a triangle

|\........| ^
|..\......| |
|....\....| x
|......\..| |
|........\| V
|.........|
|.........| 300-x
|_120 _| won't allow me to make a proper diagram but hope you guys get the picture so try to imagine it without all the dots on it.

so the equation for the hypotenuse for the lenght of wire under water will be sqrt(x^2 +120^2) = sqrt(x^2 +14400)

the equation for the lenght of wire above ground is 300-x

the function for the price for the total wire usage is P= sqrt(x^2 +14400)/15 + (300-x)/10
P=1/15 (x^2+14400)^1/2 + 1/10 (300-x)
now for the derivative of the function

P'=1/15 *1/2 (x^2 +14000)^-1/2 *2x +1/10(-1)
P'=x/(15(x^2 +14400)^1/2) - 1/10
now to find the value for x when the function equals 0

x/(15(x^2 +14400)^1/2) - 1/10=0
x/15(x^2 +14400)^1/2 = 1/10
10x=15(x^2+14400)^1/2
(10x)^2=(15(x^2+14400)^1/2)^2
100x^2=225(x^2+14400)
100x^2=225x^2 +3240000
-125x^2=3240000
x^2=3240000/-125
x^2=-25920
x=sqrt(-25920) see the problem is the negative answer so i know i have made a mistake on my approach, because i was thinking once i have obtained the value of x i could add it so the equation sqrt(x^2 +14400) and i would obtain the minimum amount of wire laid under water to minimize the cost.

Can anyone help me?? thanks!!!

Last edited: Sep 20, 2010
2. Sep 20, 2010

### Dick

If the length underwater is L meters, then the cost of laying the cable underwater is 15*L, not L/15. You've set it up so the cost of laying a meter of cable underwater is \$(1/15). That makes it cheaper to lay the cable underwater than to string in on land, so the cheapest way is to make it ALL underwater. That's why you aren't getting a critical point in a realistic place.

3. Sep 20, 2010