Calculate Lagoon Volume: x,y in km, D(x,y) = 40−160(y−f(x))2 − 40x2

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In summary, the conversation is discussing the volume of a lagoon that forms near a river as it approaches the sea. The depth of the lagoon is given by a function D(x,y) and the problem requires the use of transformation and Jacobian. The matrix used to find the transformation is 4u^2+4v^2 and the final integral is (2uv)(4u^2+4v^2)dudv. The correct inequality for D(x,y) is D(x,y)>=0.
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mikejacob33
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Please Help Soon!thanks

A river follows the path y = f(z) where x, y are in kilometers. Near the
sea, it widens into a lagoon, then narrows again at its mouth. At the point
(x, y), the depth of the lagoon is given by D(x, y) = 40−160(y−f(x))2 − 40x2 meters. The lagoon itself is described by D(x, y)  0. What is the
volume of the lagoon in cubic meters?

so far i got that this problem needs to use transformation and the jacobian.
i put x=u^2-v^2
y=2uv
find the trans using the matrix getting
4u^2+4v^2
double integral (2uv)(4u^2+4v^2)dudv
if what i did so far was right could anyone tell me how to find the limits or at least enlighten me in the right direction
 
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  • #2
I'm sorry but I have no idea what "D(x, y)  0" means. What that "box" a special symbol?
 
  • #3
sorry i wrote it down wrong
D(x,y)>=0
 

Related to Calculate Lagoon Volume: x,y in km, D(x,y) = 40−160(y−f(x))2 − 40x2

What is the purpose of calculating lagoon volume?

The purpose of calculating lagoon volume is to determine the amount of water that the lagoon can hold. This information is important for various reasons such as understanding the lagoon's capacity, planning for maintenance and restoration efforts, and assessing potential impacts on the surrounding environment.

What is the formula used to calculate lagoon volume?

The formula used to calculate lagoon volume is: V = ∫∫D(x,y) dA, where V represents volume, D(x,y) represents the depth at a given point (x,y) in the lagoon, and dA represents the infinitesimal area element. In this specific formula, the depth is represented by the equation 40−160(y−f(x))2 − 40x2.

Why is the depth of the lagoon a variable in the formula?

The depth of the lagoon is a variable in the formula because it can vary at different points throughout the lagoon. This is due to factors such as changes in topography, sedimentation, and water levels. By using a variable for depth, the formula can accurately calculate the volume of the lagoon at any given point.

How do the variables x and y relate to the lagoon volume calculation?

The variables x and y represent the coordinates of a point within the lagoon. These coordinates are used to determine the depth at that specific point, which is then used in the overall volume calculation. By using different values for x and y, the formula can calculate the volume at different points within the lagoon.

Are there any limitations to using this formula for calculating lagoon volume?

Yes, there are limitations to using this formula for calculating lagoon volume. The formula assumes a constant depth throughout the lagoon and does not account for irregular shapes or changes in depth over time. It also does not take into consideration factors such as wind and wave action, which can affect the volume of water in the lagoon. Therefore, this formula should be used as an estimate and may not provide an exact measurement of lagoon volume.

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