# Finding the derivative in an optimization problem

Hi all, I've been stuck on this question for hours and hours, I'm not sure what I'm doing wrong..

The question states,

"a new cottage is built across the river and 300 m downstream from the nearest telephone relay station. The river is 120 m wide. 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 costs to lay wire are:
-under water: $15/m -above ground:$10/m
How much wire should be laid under water to minimize cost?"

This is what I got so far:

distance of wire above ground = (300-x)/10
distance of wire under water = sqrt(x^2+120^2)/15
x= distance between the point directly across the cottage and the point where the wire crosses the river to the other side.
sqrt(x^2+120^2) = distance from the cottage to the other side of the river, aka the route of the wire to be laid under water. this is a right angle triangle so the pyth. theorem can be used to find out the hypotenuse.

The overall equation:

C= sqrt(x^2+120^2)/15 + (300-x)/10

because the problem asks for the cost, so the costs are in the denominators and distances in the numerators.

Now.. I'm not sure if i use the quotient rule or not, but I'm stuck differentiating the top half of the underwater fraction..

I've tried putting the two fractions together using a common denominator, and THEN using the quotient rule, but i still don't understand how to differentiate the top - it just makes it even more confusing.
doing that, I got:

C= [ ( 10(x^2+120^2)^0.5 + 15(300-x) )(150) + 150( 10(x^2+120^2)^0.5 + 15(300-x) ) ] / 150^2

C= ( 10(x^2+120^2)^0.5 + 15(300-x) )(150) / 22500

i'm not sure what to do next, or if i should even do it this way..
if anybody could explain it please, you could even use a different example that's similar, that would be great
i just dont understand the whole idea..

thanks a lot

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## Answers and Replies

Mark44
Mentor
Hi all, I've been stuck on this question for hours and hours, I'm not sure what I'm doing wrong..

The question states,

"a new cottage is built across the river and 300 m downstream from the nearest telephone relay station. The river is 120 m wide. 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 costs to lay wire are:
-under water: $15/m -above ground:$10/m
How much wire should be laid under water to minimize cost?"

This is what I got so far:

above ground = (300-x)/10
under water = sqrt(x^2+120^2)/15
What does "above ground" represent? How can "above ground" be equal to a number? Same comments about "under water". Try to be more descriptive and more precise about what the expressions represent. Think about what you're trying to minimize.
The overall equation:

C= sqrt(x^2+120^2)/15 + (300-x)/10

Now.. I'm not sure if i use the quotient rule or not, but I'm stuck differentiating the top half of the underwater fraction..
You shouldn't be using the quotient rule at all in this problem.
I've tried putting the two fractions together using a common denominator, and THEN using the quotient rule, but i still don't understand how to differentiate the top - it just makes it even more confusing.
doing that, I got:

C= [ ( 10(x^2+120^2)^0.5 + 15(300-x) )(150) + 150( 10(x^2+120^2)^0.5 + 15(300-x) ) ] / 150^2

C= ( 10(x^2+120^2)^0.5 + 15(300-x) )(150) / 22500

i'm not sure what to do next, or if i should even do it this way..
if anybody could explain it please, you could even use a different example that's similar, that would be great
i just dont understand the whole idea..

thanks a lot

What does "above ground" represent? How can "above ground" be equal to a number? Same comments about "under water". Try to be more descriptive and more precise about what the expressions represent. Think about what you're trying to minimize.
You shouldn't be using the quotient rule at all in this problem.

above ground represents the wire that will be laid above ground on the other side of the river, which costs $10/m under water represents the wire that will be laid under water, across the river, costs$15/m

What rule should I use for the equation?

Mark44
Mentor
How does "above ground" represent the wire -- its color, its gauge, its weight, what? Your description "above ground" is extremely vague. I'm trying to get you to think more clearly.

HallsofIvy
Science Advisor
Homework Helper
above ground represents the wire that will be laid above ground on the other side of the river, which costs $10/m under water represents the wire that will be laid under water, across the river, costs$15/m

What rule should I use for the equation?
No, they don't. "wire" is not a number so what you are calling "above ground" CAN'T be "the wire ... ". I think you mean "the distance the wire goes above ground ..."- say that.
You also start writing equations, (300- x)/10, etc. without saying what "x" represents!

Just curious where you came up with those equations. I would have used very different ones.

yup, the distance is what i meant to say, hallsofivy.

munaught - there was a similar question in my textbook, just with different numbers. I followed the steps they used for that one and got stuck trying to figure out the derivative, because the textbook does not explain that step.
and since i'm trying to figure out the cost, the cost of each distance of wire is in the denominator

here is a picture of the problem, so there is no more confusion.

http://tinypic.com/r/1zlxafo/3

could anyone please help me figure out what to do? and tell me what i'm doing wrong? i'm still not sure what rule to use to differentiate

you don't have to give me the answer, just please, walk me through getting the derivative of the equation

thank you.

edit: sorry,pic didnt show up, here's the link. http://tinypic.com/r/1zlxafo/3

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would multiplying the cost to the distance work, instead of dividing?

Cost = 15sqrt(x^2+120^2) + 10(300-x)

i still need help with the derivative

C= (15)(0.5)(x^2+120^2)^-0.5 + (10)(300-x)

Yes, you are looking for the distance and then multiplying by the cost per meter to get the total cost, not dividing.

Your equation should be set up as:

C = 15(sqrt(x2+1202) + 10(300-x)

So differentiate with respect to x, then set the derivative equal to zero and solve for x.

I wont give you an exact answer but I got a value of 100<x<120 which is perfectly reasonable. Once you have x though, remember to plug it back into the distance across the stream to get your final answer.