What is the maximum volume of an equilateral triangular prism inscribed in a sphere of radius 2?
Since the volume of an equilateral triangular prism is $\frac{\sqrt3}{4}a^2h$,where $a$ is the side length of the base triangle and $h$ is the height of the prism.How to express this volume in terms of radius of the sphere so that i can differentiate it and equate it to zero.Thanks.
Best Answer
The triangular prism takes up a certain fraction of the volume of its circumscribed cylinder. Therefore we determine first the maximal cylinder inscribed in the sphere, and bother about the equilateral triangle afterwards.
When the cylinder has height $2x$ its radius is $\rho=\sqrt{4-x^2}$, and its volume is $V_{\rm cyl}=2x\cdot\pi\rho^2=2\pi x(4-x^2)$. The latter is maximal when $4-3x^2=0$, or $x^2={4\over3}$. It follows that the maximal cylinder has $$\rho^2=4-x^2={8\over3}\ .$$ The area of the inscribed equilateral triangle then computes to $$A_\triangle=3\cdot{1\over2}\rho^2\sin(120^\circ)=2\sqrt{3}\ ,$$ so that the maximal prism obtains volume $$V_{\rm prism}=2x\cdot A_\triangle=8\ .$$