- #1

- 4

- 0

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

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

Last edited: