Math CentralQuandaries & Queries


Question from Conlan, a student:

Hi I am dong calculus at school and I'm stumped by this question:

A cone has a slant length of 30cm. Calculate the height, h, of the cone
if the volume is to be a maximum.

If anyone can help me it would be greatly appreciated.


Hi Conlan,

I drew a diagram of your cone with height h cm and r cm as the radius of the base.


The volume of the cone in cubic centimeters is given by

\[V = \frac13 \pi\; r^2 h.\]

To complete the setup of this problem you need an equation relating $r$ and $h$ which you can use to eliminate $r$ from the equation for $V$ and allow you to use the calculus you know to maximize $V.$ What does Pythagoras Theorem tell you about the triangle ABC?


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