Torricelli's Law Problem: ellipsoid (x/5)^2 + (y/4)^2 + (z/3)^2 = 1

Tom999

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Problem: The inside of a container is shaped by an ellipsoid (x/5)^2 + (y/4)^2 + (z/3)^2 = 1. A draining at (x,y,x) = (0,0,-3) and an air intake at (x,y,z) = (0,0,3) are made to allow a draining constant k = 8.8. Compute the time to empty the container that's fully filled initially.

My reasoning: (A(x,y))dz/dt = -8.8rad(z) should be the setup. I need the area of vertical elliptic cross-sections, each with area xy(pi) with x and y increasing as the fluid approaches the halfway point and decreasing as the rest of the container drains. Once I have the area function I can solve the separable DE. The issue is that I don't know where to begin in expressing the area as a function of z.

Thanks
 
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