Calculating the Depth of an Artificial Lake in a Steep Valley

[Omega_Prime]

"Building an artificial lake"

Howdy PF this is actually a Calc III project I'm working on with two others. We are in the beginning stage and are having a little trouble on how to exactly go about doing this problem...

Homework Statement

"A small man made lake is being proposed on a stretch of river running through a steep valley. Surveyors have determined the elevation of the land at points situated on a [rectangular] grid."

The proposed lake shall have a capacity of 5 billion cubic feet.

Do not use parallelepipeds to calculate volume.

Homework Equations

V = $$\int$$$$\int$$ z(x,y) dA

The Attempt at a Solution

Ok, so I'm given the elevation of said stretch of river/land at intervals of 500 ft. The goal is to stick a dam somewhere and create an artificial lake with a volume of 5 billion cubic feet. We know the ground (elevation) is not perfectly flat, so the depth will not be constant throughout. However, I have to find some value (h) where h will be the elevation of the water level for this lake.

I can see the lake-floor changing depending on its position (x,y). For h to remain constant for all points (x,y) the depth will have to change accordingly (ie depth = h - z(x,y)). This is kinda where I'm stuck...

In the table I'm given (much too large for me to include, but if you can imagine, a 3D graph of the data points creates a messy and jagged parabolic cylinder along the x-axis) if I go horizontally across (the data table) my y values are constant while my x and z change. If I go vertically down a column my x is constant while y and z change. Should I use partial derivatives to define a plane over a triangular domain for every 3 data points? If so, I'll have a chunk of volume for each triangle. Then it's just a matter of solving a huge simultaneous equation for h? I feel so stupid right now...

I'm really struggling with this... I'm sure my post is a little confusing itself. Has anyone worked with a particular problem like this? If you need me to clarify or expand upon anything please feel free to ask - I hope this type of homework assignment is acceptable for this section.

 

[Stephen Tashi]

I think what you should do depends on the purpose of the project and whether you are expected to write or use a computer program as opposed to being restricted to using a calculator or only hand calculations.

Your idea of using triangular regions is good. I don't know whether there is any real world evidence that using curved surface fitted by partial derivatives is any better than simply using the plane itself to define the boundary of the volume. However, you have to have keep in my that as h is increased, some new new triangular regions will begin to hold water that didn't before. Imagining real world data, I don't see any way to set up a simple set of equations to solve for h because there must be some logic that says "if h < h_L, the lowest vertex of the triangle, this region makes no contribution to the volume, other wise it contributes such-and-such a function f(h) till h rises above the highest vertex h_U of the triangle. After that it contributes f(h_U) + k h.

Figuring out f(h) looks like an exercise that involves symbolic calculation. A computer program could solve the problem numerically by gradually increasing h till the desired volume is reached or doing some simple equation solving technique like "bisection".

 

[Omega_Prime]

I think what you should do depends on the purpose of the project and whether you are expected to write or use a computer program as opposed to being restricted to using a calculator or only hand calculations.

Your idea of using triangular regions is good. I don't know whether there is any real world evidence that using curved surface fitted by partial derivatives is any better than simply using the plane itself to define the boundary of the volume. However, you have to have keep in my that as h is increased, some new new triangular regions will begin to hold water that didn't before. Imagining real world data, I don't see any way to set up a simple set of equations to solve for h because there must be some logic that says "if h < h_L, the lowest vertex of the triangle, this region makes no contribution to the volume, other wise it contributes such-and-such a function f(h) till h rises above the highest vertex h_U of the triangle. After that it contributes f(h_U) + k h.

Figuring out f(h) looks like an exercise that involves symbolic calculation. A computer program could solve the problem numerically by gradually increasing h till the desired volume is reached or doing some simple equation solving technique like "bisection".

Thanks Stephen, I had briefly thought about the rising water level affecting the footprint underneath and thus affect integration. We're using MS Excel for most of the grunt work, I'm now thinking I may have to come up with a decent over/under estimation using different shapes (like a parallelepiped) and go from there.

 

[Stephen Tashi]

Based on my limited experience with Excel, it is possible to put a function with somewhat complicated if-then logic in a cell to compute the value of that cell.

Another real world complication is the possibility of lakes-within-islands. You could have a depression in the Earth lower than h, but it might be surrounded by a rim of higher terrain. Perhaps if you examine the data visually you can detect if this could occur.