Donut & Tea

Calculus Level 4

A donut with hollow circle of radius $r$ and cross section circle of radius $r$ is shaped along with 3 copies of series of cuboids with height $2r$ and square-bases with decreasing side of $\dfrac{2r}{n}$ , where $n$ is a positive integer.

As $n$ approaches infinity, the cross section from the top will tend to a T-like base as shown above.

Which figure would have more volume as $n$ reaches infinity?

T-structure Both would have the same volume Donut

1 solution

Worranat Pakornrat
Dec 30, 2016

The volume of the donut = volume of torus with cross section of $r$ and major radius of $2r$ (distance from donut's center to cross section's center) = $(\pi r^2 )(2\pi(2r)) = 4\pi^2 r^3$ .

The volume of the T-structure = $3(2r)\sum \dfrac{4r^2}{n^2} = 24r^3\sum \dfrac{1}{n^2}$

According to Basel's theorem ,the sum of reciprocal squares equals to $\dfrac{\pi^2}{6}$ .

Thus, the T-structure's volume = $24r^2 \dfrac{\pi^2}{6} = 4\pi^2 r^3$ .

So both figures have the same volume.

Donut & Tea

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