zmudz
  • zmudz
Find the largest possible volume of a cone that fits inside a sphere of radius one.
Mathematics
schrodinger
  • schrodinger
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anonymous
  • anonymous
Hmm, are we talking about a regular cone, or a slanted one? I'll assume the first case (just because I imagine it's much easier). I wouldn't be surprised if there's a very quick way to do this, but this is certainly one way to model the situation. Consider the upper half of the circle centered at the center with radius \(1\), i.e. defined by the curve \(y=\sqrt{1-x^2}\). It's fairly obvious that for any cone inscribed within the sphere, the maximum volume of the cone can only be attained if the vertex of the cone is fixed at some point on the sphere. The slant height of the cone can then be represented by a line through this point on the sphere and another. The idea here is to use the half-circle and this line as a cross-section: |dw:1441845241839:dw|
anonymous
  • anonymous
Recall the volume of a cone: \[V=\frac{\pi}{3}(\text{radius of base})^2(\text{height})\] which translates to \[\begin{align*} V(a)&=\frac{\pi}{3}\left(\sqrt{1-a^2}\right)^2(1-a)\\[1ex] &=\frac{\pi}{3}(1-a^2)(1-a) \end{align*}\] You can then proceed with the the derivative test for extrema to show that \(a=\dfrac{1}{3}\) gives the maximum volume of the cone.

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