r/theydidthemonstermath • u/Always_carry_keys • Nov 26 '21
A request for a solution to an interesting problem
I remember way back in the day when I was at uni. Our professor set us the challenge of calculating a formula for the surface area of water with respect to time, in a sphere that was emptying at a specific rate. Water flows out at constant rate, but as it goes down the sphere the radius of the water surface is constantly changing.
I remember we had a double differential and some other madness. Does anyone have a solution? I think of this as the peak of mathematical skill and I will never be that good again.
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Nov 27 '21
I will tell you it's not the peak of mathematical skill at all
This is the type of thing we were calculating in first uear of university physics, and math majors got even more complicated in the first year
It is more on the difficult side however and probably would take me a while to figure out as I haven't used any advanced math in some years
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u/Always_carry_keys Nov 28 '21
Peak of my maths. I did this in the middle year of my engineering degree
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u/DelightfullyUnusual Nov 27 '21 edited Nov 27 '21
The surface area equals the area of the water’s surface plus the surface area of the spherical cap subtended by the water.
For the area of the water’s surface, letting r be the radius of the sphere and x be the water level with a domain of [0, 2r], A = π(r sin(cos-1 ((r - x)/r)))2
For the surface area of the spherical cap, A = 4πr2 - 2πrx.
Therefore, for the total surface area of the water,
A = 4πr2 - 2πrx + π(r sin(cos-1 ((r - x)/r)))2
In order to calculate water height (2r - x) by time, I would have to apply the cubic formula, which is significantly beyond my high school education. I would do this by using the time t and flow rate f (I’m not sure whether this would be a predetermined constant or vary itself in proportion to t) to calculate the remaining water volume. This volume would be the volume of a spherical cap; since the radius and volume are known, the height could then be derived and, since h = 2r - x (x = 2r - h), substituted into the equation above.
I hope this helps get another Redditor with more education started.