Name: Local Maxima and Minima of a Function
Uploaded: 2019-07-19
Duration: 14 min 17 s
Description: Do you know what is minima or maxima of a function? This video also explains how to find local maxima and minima by differentiation in a detailed manner.

Question

Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to

\frac{2}{3}

of the diameter of the sphere.

Accepted Answer

Let $h, r$ and $R$ be the height, radius of base of the cone and radius of the sphere, respectively.

Then,

$h = R + x$

$h = R + \sqrt{R^{2} - r^{2}}$

$\Rightarrow (h - R)^{2} = R^{2} - r^{2}$

$\Rightarrow h^{2} + R^{2} - 2 hr = R^{2} - r^{2}$

$\Rightarrow r^{2} = 2 hR - h^{2} \dots (1)$

Volume of cone $= \frac{1}{3} {πr}^{2} h$

$\Rightarrow V = \frac{1}{3} πh (2 hR - h^{2})$ [ From eq. $(1)$ ]

$\Rightarrow V = \frac{1}{3} π (2 h^{2} R - h^{3})$

$\Rightarrow \frac{dV}{dh} = \frac{π}{3} (4 hR - 3 h^{2})$

For maximum or minimum values of $V$ , we must have

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

$\Rightarrow \frac{π}{3} (4 hR - 3 h^{2}) = 0$

$\Rightarrow 4 hR = 3 h^{2}$

$\Rightarrow h = \frac{4 R}{3}$

Now,

$\frac{d^{2} V}{{dh}^{2}} = \frac{π}{3} (4 R - 6 h)$

$\Rightarrow \frac{d^{2} V}{{dh}^{2}} = \frac{π}{3} (4 R - 6 \times \frac{4 R}{3})$

$\Rightarrow \frac{d^{2} V}{{dh}^{2}} = \frac{- 4 πR}{3} < 0$
So, the volume is maximum when $h = \frac{4 R}{3}$

$\Rightarrow h = \frac{2}{3}$ (Diameter of sphere)

Hence proved.

Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to $\frac{2}{3}$ of the diameter of the sphere.

Important Questions on Maxima and Minima

Learn and Prepare for any exam you want !

Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to 23 of the diameter of the sphere.

Important Questions on Maxima and Minima

Learn and Prepare for any exam you want !

Show that the cone of the greatest volume which can be inscribed in a given sphere has an altitude equal to $\frac{2}{3}$ of the diameter of the sphere.