Radical Problem

[tlwaring]

Here's a problem that is driving me crazy. Can someone help?

A natural gas line running along a river is to be connected from point A to a cabin on the other bank located at point D. The width of the river is 500 feet, and the distance from point A to point C is 1000. The cost of running a pipe along the shoreline is $30 per foot, and the cost of running it underwater is $50 per foot. The cost of connecting the gas line from A to D is to be minimized.

A. Write an expression that gives the cost of running the line from A to B if the distance between these points is x feet.

B. Find the distance from B to D in terms of x.

C. Write and expression that gives the cost of running the line from B to D.

D. Use your answer from parts A and C to write an expression that gives the cost of running the line from A to B to D.

E. Use your answer in part D along with trial and error to obtain a "reasonable" minimum cost estimate.

This is one reason I can't stand story problems. Help! :

 

[galactus]

What are points B and C?. Your posts seems a little confusing.

 

[Gene]

Before we start scribling equations:
Is C straight across the river from A?
Is B on the river shore where the pipe enters the water?

 

[tlwaring]

Point B is between Point A & C on the river bank across from the cabin.

C is across the river from D.

Thanks for your help!

 

[Gene]

We can be more helpful if you show how far you have gotten.
--------------------
Gene

 

[galactus]

Is this what you mean?.

We'll see what the others have to say, but here is one way to look at it assuming the diagram is correct:

The portion under water is $50 per foot and the portion on land is $30 per foot. Using ol' Pythagoras we get

50(500²+x²)^1/2+30(1000-x).

Now differentiate and get (50x/(sqrt(x²+250000)))-30.

Set to 0 and solve for x and get x=375.

sqrt(500²+375²)*50=31250

30(1000-375)=30*625=18750

18750+31250=$50,000

 

[Gene]

What I had in mind is:
I'm sure you can do A)
I think you can do B) and with that I'm sure you can do C)
...
In other words, where are you getting stuck?

 

[tlwaring]

The diagram is correct. The problem that I am having with it is after working within a group, no one can come up with one direction to follow. We have everything from C=30x to x=square root of 5. There were even costs of over $1 million. We are trying to make heads or tails of it but we seem to making more problems. Help!

 

[Gene]

Back again. I should have looked at the problem earlier. I've been playing with it but in looking at your diagram it appears you have a distance between C & D. I can't make out what it is and it isn't part of the written problem. If you do know what it is, I can help. I've been trying it with that as a variable but without any joy.

 

[tlwaring]

The distance between C & D is 500 ft. Believe me, we haven't had much joy with it either.

Thank you so much for your help.

 

[Gene]

tlwaring:
My diagram is:

D
|\ z
| \
|  B
|  }..
|w { .
|  } .. 
| y{  . x
|  }  ..
|  {   .
|  }   ..
C       A

The .s are the line x, steeper than DB. { is the river bank, y long. | is the other bank, CD and is w long.

From the right triangles
x=sqrt(y²+500²)
z=sqrt((w-y)²+500²)

Cost = 50z+30x

Solving for y
y=sqrt(x²-500²)

Getting cost into terms of x:

Substituting for z
C=50*sqrt((w-y)²+500²) + 30x

Substituting for y
C=50*sqrt((w-sqrt(x²-500²))²+500²) + 30x

That is as far as I can take it without confirmation that you know w. The diagram cannot be correct. If it were there is no sense in E 'cause the way to minimize cost is to move B up to take advantage of the cheaper rate of land digging.

 

[Gene]

We overlapped. I'm a slow typist.
I get x=597.1 feet and z=529.3 feet, cost = $44,378 with w = 500 and my diagram.

PS. Nevermind. I now, on a carefull rereading, see that I had my river going the wrong way.
---------------
Gene

 

[tlwaring]

Gene -

Thanks so much for all your help. Believe me, it gives us a path to follow!