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Omega_Prime
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"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...
"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.
V = [tex]\int[/tex][tex]\int[/tex] z(x,y) dA
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.
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 = [tex]\int[/tex][tex]\int[/tex] 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.