# Thread: maximum volume that can fit in a cone...

1. ## maximum volume that can fit in a cone...

Here is a calculus problem, I've been trying to solve but I haven't been able to, please help me! thanks

A right circular cone has a height of 10 and a radius of 3. find the dimensions of the cylinder with maximum volume that can fit inside this cone such that the base of the cylinder lies on the base of the cone. Also give maximum volume and verify that the volume is a maximum.

2. Originally Posted by panywff
Here is a calculus problem, I've been trying to solve but I haven't been able to, please help me! thanks

A right circular cone has a height of 10 and a radius of 3. find the dimensions of the cylinder with maximum volume that can fit inside this cone such that the base of the cylinder lies on the base of the cone. Also give maximum volume and verify that the volume is a maximum.
See attachment.

The volume of a cylinder is calculated by:

$\displaystyle V_{cyl} = \pi \cdot r^2 \cdot h$

According to my sketch you can use similar triangles.

$\displaystyle \dfrac h{R-r} = \dfrac HR ~\implies~ h = \dfrac{H(R-r)}{R}$

Substitute this term into the first equation:

$\displaystyle V(r)=\pi \cdot r^2 \cdot \dfrac{H(R-r)}{R} = -\pi \frac HR \cdot r^3 + \pi H\cdot r^2$

Calculate V'(r) and solve V'(r) = 0 for r.

I've got $\displaystyle r = 0~\vee~r = \frac23 R$

Since V''(r) < 0 the volume must be a maximum.

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### max.size cube fit in cylinder

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