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Given a cone of height $h$ and an angle $\theta$ between the axis and it's lateral height. The cone is filled with water. A sphere of radius $R$ will be inserted into the cone. Find the value of $R$ which maximizes the water overflow.

This one seems tricky to me. I tried to create a "overflown volume function" $f(r)$ and then maximize it. However, a function $f(r) = V_{cone} - V_{sphere}$ (like i did) ignores the fact that the maximum overflow can happen in a case which the sphere does not fully enter in the cone, so i'd have to consider that, but i don't know how. Or maybe the maximum volume will always be when the entire sphere is inside the cone? I can't visualize that. Thanks.

How can i model a function that considers the fact that the sphere may not fully enter in the cone?

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For any given $h$ and $\theta$, there is a critical radius $R=R_0$ at which the sphere is tangent not only to the curved side of the cone but also to the surface of the water at the top of the cone.

The sphere with radius $R=R_0$ displaces more water than any smaller sphere, so clearly the displacement is maximized for some $R \geq R_0.$

There is a larger spherical radius, $R=R_1$, at which the sphere is tangent to the cone but the circle of tangency is exactly at the surface of the water. If $R > R_1$ then we have a sphere that touches the cone only at the top edge of the cone; the part of the sphere inside the cone is a subset of the part of the sphere of radius $R=R_1$ inside the cone, so the volume of water displaced is maximized for some $R\leq R_1$.

The volume of water displaced is then the volume of the sphere minus the volume of whatever part of the sphere is above the plane of the surface of the water. If the highest point on the sphere is at a distance $d$ above that plane, the part of the sphere above that plane is a spherical cap of height $d$ from a sphere of radius $R,$ which has a volume $$ \frac13 \pi d^2(3R-d). $$

Possibly you were meant to work out that formula for yourself using a disk method of integration.

Alternatively, let $d$ be the distance from the lowest point of the sphere to the plane at the top of the cone, and then the volume of water displaced is the volume of a cap of height $d$. (Either formula works just as well for $R < d \leq 2R$ as for $0\leq d\leq R.$)

In any case, you can assume $R_0 \leq R \leq R_1$, that is, the sphere is tangent to the curved surface of the cone and also intersects the plane at the top of the cone. Then $d$ is a function of $h,$ $\theta,$ and $R.$ Treating $h$ and $\theta$ as constants and $R$ as a variable, the volume of the part of the sphere within the cone is a function of $R$ that you can maximize.

To complete the solution, you should confirm that the answer is in the range where your formulas are valid, that is, $R_0 \leq R \leq R_1.$

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A visualisation of a problem instance:

small sphere in cone (Large version)

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