Application of differentiation question (1 Viewer)

nazfiz · Oct 3, 2011

The slant edge of a right circular cone is 6cm in length. Find the height of the cone when the volume is a maximum.

Thanks in advance (working out would be appreciated)

SpiralFlex · Oct 3, 2011

nazfiz said:
The slant edge of a right circular cone is 6cm in length. Find the height of the cone when the volume is a maximum.

Thanks in advance (working out would be appreciated)

Let's draw a pretty diagram,

Let the height of the cone by $h$ and let the radius of the cone be $r$ .

Look at the inside triangle,

$r^2+h^2=36$

$\therefore r^2=36-h^2$

We know that the volume of a cone is,

$V=\frac{1}{3}\pi (36-h^2)h$

$\therefore V =12\pi h-\frac{1}{3}h^3\pi$

$\frac{dV}{dh}=12\pi-h^2\pi$

For maximum/minimum,

$\frac{dV}{dh}=0$

$\therefore 12\pi-h^2\pi=0$

$h=2\sqrt{3}$

Testing for maximum/minimum,

$\frac{d^2V}{dh^2}=-2\pi h$

At $h=2\sqrt{3}$

$\frac{d^2V}{dh^2}=-2\pi(2\sqrt{3})<0$

Hence the volume is maximised when the height of the cone is $2\sqrt{3}\:\textup{cm}$

nazfiz · Oct 3, 2011

Thanks man. Spiralflex you are clear cut awesome. Repping you.

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Application of differentiation question (1 Viewer)

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