Question

Among all circular cones with a slant height of 15, what are the radius and height that maximize the volume of a cone?

Answer 1

We need two formulas Volume of circular cone and pythagorean theorem,

\[c^{2}=a^{2}+b^{2}\]

we know 15 is the hypotenuse from the problem so
\[c^{2}=15\]

for a,b........one of them will be the radius and the other the height....which one will be which? Makes no difference, pick whatever you like. i will choose the following
\[a=radius\]
\[b=height\]

Volume formula for circular cones,

\[V_{cone}=\pi(radius)^{2}(height)\]

To optimize we just need to write the volume formula in terms of one variable only....so, either write your "height" in terms of your radius OR your "radius" in terms of your height....makes no difference which way you choose, the derivative won't care. I will choose the following
\[c^{2}=a^{2}+b^{2}\]
\[15=(radius)^{2}+(height)^{2} \;\;\;\;\;\; Substitutions\]
\[height=\sqrt{15-(radius)^2} \]

So, now substitute the height in terms of the radius back into the Volume formula

\[V_{cone}=\pi(radius)^{2}(\sqrt{15-(radius)^2})\]

differentiate, set equal to zero find the radius and then use pythagoras(above) to dinf the height height.