Name: What are Asymptotes?
Uploaded: 2019-07-19
Duration: 31 min 35 s
Description: In this video, we will learn about asymptotes and their definition, formulas, slant asymptotes, asymptotes of different graphs such as hyperbola etc.

Question

Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius

' a'

is

\frac{8}{27}

(volume of the sphere).

Accepted Answer

Given: sphere of radius is $a$

Let $x$ be the radius of the inscribed right circular cone and $h$ be its height, then

${OM}^{2} + {MC}^{2} = {OC}^{2}$ $[$ From Pythagoras theorem $]$

$\Rightarrow (h - a)^{2} + x^{2} = a^{2}$

$\Rightarrow x^{2} = 2 a h - h^{2} . . . . . . . . . (1)$

If $V$ is the volume of the inscribed cone, then

$V = \frac{1}{3} π x^{2} h = \frac{1}{3} π (2 a h - h^{2}) h = \frac{π}{3} (2 a h^{2} - h^{3})$ $[∵ x^{2} = 2 a h - h^{2}]$

Differentiating w.r.t. $h,$ we get

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

Now for maximum volume, $\frac{d V}{d h} = 0 \Rightarrow \frac{π}{3} (4 a h - 3 h^{2}) = 0$

$\Rightarrow h (4 a - 3 h) = 0 \Rightarrow h = 0, \frac{4 a}{3}$ but $h > 0 \Rightarrow h = \frac{4 a}{3}$

Also, for $h = \frac{4 a}{3}, \frac{d^{2} V}{d h^{2}} = \frac{π}{3} (4 a - 6 \cdot \frac{4 a}{3}) = - \frac{4}{3} π a < 0$

$\Rightarrow V$ is maximum when $h = \frac{4 a}{3}$

Therefore, maximum volume $= \frac{π}{3} h^{2} (2 a - h) = \frac{π}{3} (\frac{4 a}{3})^{2} (2 a - \frac{4 a}{3}) = \frac{π}{3} \cdot \frac{16 a^{2}}{9} \cdot \frac{2 a}{3}$

$= \frac{8}{27} \cdot \frac{4}{3} π a^{3} = \frac{8}{27}$ of the volume of sphere. $[∵$ Volume of sphere of radius $a = \frac{4}{3} π a^{3}]$ .

Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius $' a'$ is $\frac{8}{27}$ (volume of the sphere).

Important Questions on Applications of Differential Calculus

Learn and Prepare for any exam you want !

Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius 'a' is 827 (volume of the sphere).

Important Questions on Applications of Differential Calculus

Learn and Prepare for any exam you want !

Show that the volume of the largest right circular cone that can be inscribed in a sphere of radius $' a'$ is $\frac{8}{27}$ (volume of the sphere).