Name: Global Maxima and Minima
Uploaded: 2019-07-19
Duration: 4 min 39 s
Description: There is a difference between local and global products. Watch this video to know more about global maxima and minima and local maxima and minima.

Question

Prove that volume of largest cone, which can be inscribed in a sphere, is

\frac{8}{27}

part of sphere.

Accepted Answer

Cone of largest volume inscribed in a sphere of radius $R$

Volume of sphere $= \frac{4}{3} π R^{3}$

Let $O C = x$

Radius of cone $= B C$

Height of cone $= h = A C = R + x$

In right-angled triangle $B O C$ , $O B^{2} = B C^{2} + O C^{2}$

$\Rightarrow B C = \sqrt{R^{2} - x^{2}}$

Let $V$ be the volume of cone.

$\Rightarrow V = \frac{1}{3} π r^{2} h$

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

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

Differentiating above equation with respect to $x$ , we have

$\frac{d V}{d x} = \frac{π}{3} (R^{2} - 2 R x - 3 x^{2}) . . . . . (2)$

Put $\frac{d V}{d x} = 0$

$\Rightarrow R^{2} - 2 R x - 3 x^{2} = 0$

$\Rightarrow (R + x) (R - 3 x) = 0$

$∴ x = \frac{R}{3}$ (Since radius cannot be negative)

Now differentiating equation $(2)$ with respect to $x$ , we have

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

$\Rightarrow \frac{d^{2} V}{d x^{2}} = - \frac{π}{3} (2 R + 6 \frac{R}{3}) = - \frac{4 π R}{3} < 0$

Thus, Volume will be maximum for $x = \frac{R}{3}$

Therefore,

Substituting the value of $x$ in equation $(1)$ , we have

$V = \frac{π}{3} (R^{2} - \frac{R^{2}}{9}) (R + \frac{R}{3}) = \frac{π}{3} \times \frac{32 R^{3}}{27} = \frac{8}{27} (\frac{4}{3} π R^{3})$

Hence, proved.

Prove that volume of largest cone, which can be inscribed in a sphere, is $\frac{8}{27}$ part of sphere.

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Prove that volume of largest cone, which can be inscribed in a sphere, is 827 part of sphere.

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Prove that volume of largest cone, which can be inscribed in a sphere, is $\frac{8}{27}$ part of sphere.