A container of parabolic cross-section is made by rotating the curve about the y-axis (a is a constant).?

Question

The container is filled with water to level h. What is the volume of water in the container (in terms of h and a)?
What is the area of the surface of the water when filled to level h (in terms of h and a)?
The water is evaporating from the container at a rate proportionate to the surface area, with proportionality constant k. Approximate the time t in which the water level drops from to level h (in terms of h, Δh, a, and k). Assume that Δh is small, and that the rate doesn’t change much during this time.
If the water is initially filled to level H, how long will it take to completely evaporate?
I need to show all my work thanks

Answer 1

If the container is made by rotating a curve y = f(x) around the Y axis, then for each x, the cross-section at y = f(x) is either a circle or a set of rings.

Assume that it is a circle (you can generalize to rings, but in this case you don't need to), then we can define a function g(y) which is the inverse of f(x). That is:

g(f(x)) = x and f(g(y)) = y

Then since h = y, g(h) is the radius of the circle at y = h.

Obviously, this lets you solve the area (pi * r^2) and simple evaporation parts of the problem.

Volume is the integral of the area from y = 0 to y = h.

Total evaporation time can be done with integration as well, but a shortcut is to note that since evaporation is proportional to surface area, total time for evaporation is just proportional to the volume.

Answer 2

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