I need help with this calculus problem?

Question

This is not for a grade, this for my job. I'm testing the content at my company website,since students are saying there answers are being marked wrong when they're correct. Thanks I appreciate it.


A cable is to be run from a power plant on one side of a river 2,400 meters wide to
a factory on the other side, 2,600 meters downstream. The cost of running the cable
under the water is $55 per meter, while the cost over land is $44 per meter. What is
the most economical route over which to run the cable? Round your answer to the
nearest integer.

Oops I'm sorry forgot the second part of the question. Here it is:

The most economical route way is to run a cable under the water___________ meters and______ meters over the land.

Answer 1

let underwater route be hypotenuse of rt triangle with legs 2400 and x, leaving 2600-x for land portion of route. cost is
C = 44(2600-x) + 55√(2400² + x²)
C = 44(2600) - 44x + 55√(2400² + x²)
C' = -44x + 55/[2√(2400² + x²)] • 2x
C' = -44x + 55x/√(2400² + x²)
min when C' = 0, so
44x = 55x/√(2400² + x²)
4x√(2400² + x²) = 5x
16x²(2400² + x²) = 25x²
dividing by 16x², which makes x=0 one solution,
2400² + x² = (5/4)²
which of course has no real solution, so
best route is 2600 m on land, then straight across 2400 m under water.

That's the naive solution, and it's completely wrong. At x = 0, cost is at its max, $246,400, and at x = 2600 (route all underwater), cost is at its min, $194,610. The slope of the tangent to the curve is never 0 on the interval [0,2600]. Derivative = 0 gives you relative min or MAX. You have to test the endpoints of the interval before you know what's going on.

Answer 2

Calculus schmaculus.

C = 55*((2600-x)^2+2400^2)^.5+44*x

By inspection minimum Cost C when x =0.
(Graph shows monotonic function.)

Run underwater from power plant to factory,
3538 M under water, 0 M over land.

Answer 3

f (x) = x^3 - 12x + one million . . . the 1st by-product set to 0 reveals turning or table certain factors f ' (x) = 3x^2 - 12 3x^2 - 12 = 0 3 * (x + 2) * (x - 2) = 0 x = 2 ... x = - 2 . . . the 2d by-product evaluated at x = 2 and -2 determines if those factors are min, max, or neither. f ' ' (x) = 6x f ' ' (2) = 6*2 = 12 <== effective fee shows x=2 is an area minimum f ' ' (-2) = 6*(-2) = -12 <== damaging fee shows x=-2 is an area optimum a.) x = - 2 is a optimum, and x=2 is a minimum ... so x = - infinity to -2 is increasing x = -2 to +2 is lowering x = +2 to + infinity is increasing b.) f (-2) = (-2)^3 - 12*(-2) + one million = 17 f (2) = (2)^3 - 12*(2) + one million = - 15 c.) . . . the 2d by-product set to 0 reveals inflection factors, or the place concavity variations 6x = 0 x = 0 <=== inflection factor x = - 2 is a optimum, so could desire to be concave down concavity variations on the inflection factor(s) ... so x = - infinity to 0 is concave down x = 0 to + infinity is concave up

Answer 4

OK I suppose the diagram as follows
http://i24.tinypic.com/1266920.jpg
the function of the cost as in my diagram is
C= X*55 + 44*(1000 - sqr(X^2 -2400^2))
to make the cost to the minimum so get C'=0

C'= 0 = 55 -44X/sqr(X^2-2400^2)
so 55*sqr(X^2-2400^2)=44x
3025(x^2 - 5760000)=1936 x^2
x^2= 17424000000/1089
x^2 =16000000
x= 4000 >2600
so the most economic is to run all in the water
the cost will be
2600*55= 143,000 $

as a test try 1 meter less
i.e. 2599 in water
the cost will be
143059 > 143000
********************************
but if the diagram is as follows
http://i20.tinypic.com/j5v3nq.jpg
so the cost will be
C=55*x + 44(2600- sqr(x^2 -2400^2))
C' is the same
so x=4000 >3539
so the cost will be
55*3539= 194645 $

Answer 5

This is a right triangle with legs 2400 and 2600, so by Pythagorean Theorem the hypotenuse is 3538.36.

If the cable runs along the hypotenuse underwater it costs 3538.36*55 = 194610.

If it runs along the legs, it runs 2600 m on land and 2400 m on water at cost =2600*45+2400*55 = 249000.

So choose first way.

Answer 6

Every time I worked it I came up with the most economical answer to be to run it under the water the whole way.
(2600)^2 + (2400)^2 = (3538.36)^2

3538.36 * 55 = $194609.87

Answer 7

you dont say how deep it is to the river bed or if that is where the cable will lie or beneath the river bed nor how far the factory or the power plant is from the river bank, with the information given you'll need 3539 meters of cable all to go underwater @ $55 per metre

Answer 8

Underwater costs $194,610 whereas over land is $220.000. based on the assumption that we have a right angle triangle

Answer 9

As pythagorus. This is his schtick.