You have a cube of volume $\ell$ cubic meters, and a machine that takes the following:
- Input $S \subset \mathbb R^3$, where $S = \bigcup_{\text{selected}} f_\text{selected}(g_\text{selected})$, where each $f_i$ is a distance and angle preserving injection between an object in your inventory and $\mathbb R^3$, and $g_i$ is an object in your inventory. It essentially allows you move around your objects in space, but not stretch them before inputting.
- If for a given $\alpha>0$, $S$ is $\epsilon$ close to a cube of side length $\alpha$, meaning that only $\epsilon$ mass is missing from the cube of side length $\alpha$ when intersected with $S$, AND only $\epsilon$ mass is extra from the cube of side length $\alpha$ when intersected with $S$, then the machine consumes all your items $S$, and gives you back the following:
- A cube of side length $\alpha$ where one edge is contracted
- A cube of side length $\alpha'$, where $\alpha'$ is such that the total volume of $S$ and the total volume of the two outputs of the machine are equal.
You may use the machine as many times as you wish, and then you must pack all of your items into a sphere of minimal radius. Given $\ell$, denote the strategy giving the smallest possible radius as $r(\ell, \epsilon)$.
Define the map $f : \mathbb R^{\ge0} \to \mathbb R$ as $x \mapsto r(x,\sin x)$.
Q: Determine $\int_0^{1,000} f(t) dt$.
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Right, so I have a few thoughts on this problem. For one, when $\epsilon=0$, the machine will only accept perfect cubes, so you will only be able to activate it once to receive a one-edge contracted cube, and a smaller cube. Suppose that the volume of a edge-contracted cube of sidelength $\ell$ is $\gamma \ell^3$ for any $\ell$, since you will always just have a constant times the normal volume. So essentially the $\epsilon$ value will determine how many times you can do this operation, and the operation will always help decrease the radius of the packing sphere (just noticed this by manual manipulation of the objects). However, one thing I'm unsure of, is whether or not it is actually possible to abuse the machine in the following way. Could you potentially have a large enough cube and a small enough epsilon to make a decreasing-volume series of edge-contracted cubes, and then stack them in such a way that they are at most $\epsilon$ away from being a cube, so that the machine would give a smaller contracted-cube than the largest you started with, meaning you would be able to FURTHER decrease the radius of a sphere in that kind of setup. That is something I am more unsure is possible, and I can't imagine a way of actually figuring out whether or not that would be possible with the parameters for the integral. Any help on how one might even approach a problem like this would be useful. Thanks
Edit: Also, clearly $f(0)=0$, at least that is clear