Name: Concavity and Point of Inflection of a Function
Uploaded: 2019-07-19
Duration: 23 min 41 s
Description: This video talks about concavity and the point of inflection of a function with sketching a graph. Let's learn one of the most important applications of deri...

Question

Find the altitude of a right circular cylinder of maximum volume inscribed in a sphere of radius

r

.

Accepted Answer

Given, radius of sphere $= r$ .

Let, $h$ be the height and $x$ be the diameter of cylinder.

In $∆ A B C$ , using Pythagoras theorem, $(C B)^{2} + (A B)^{2} = (A C)^{2}$

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

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

$\Rightarrow h^{2} + x^{2} = 4 r^{2}$

$\Rightarrow x^{2} = 4 r^{2} - h^{2}$ ...(i)

To find maximum volume of cylinder.

Let $V$ be the volume of cylinder,

$V = π$ $($ radius $)^{2}$ $\times$ $($ height $)$

$\Rightarrow V = π {(\frac{x}{2})}^{2} \times h$

$\Rightarrow V = π \times \frac{x^{2}}{4} \times h$

$\Rightarrow V = π \frac{(4 r^{2} - h^{2})}{4} \times h$ [from(i)]

$\Rightarrow V = \frac{4 r^{2} π h}{4} - \frac{π h^{3}}{4}$

$\Rightarrow V = π h r^{2} - \frac{π h^{3}}{4}$

Differentiating w.r.t $h$ ,

$\Rightarrow \frac{d V}{d h} = \frac{d (π h r^{2} - \frac{π h^{3}}{4})}{d h}$

$\Rightarrow \frac{d V}{d h} = π r^{2} \frac{d (h)}{d h} - \frac{π}{4} \frac{d (h^{3})}{d h}$

$\Rightarrow \frac{d V}{d h} = π r^{2} - \frac{π}{4} (3 h^{2})$

$\Rightarrow \frac{d V}{d h} = π r^{2} - \frac{3 π}{4} h^{2}$

Putting $\frac{d V}{d h} = 0$

$\Rightarrow π r^{2} - \frac{3}{4} π h^{2} = 0$

$\Rightarrow \frac{3}{4} π h^{2} = π r^{2}$

$\Rightarrow h^{2} = \frac{π r^{2}}{\frac{3}{4} π}$

$\Rightarrow h^{2} = \frac{4 r^{2}}{3}$

$\Rightarrow h = \frac{2 r}{\sqrt{3}}$

Finding $\frac{d^{2} V}{d h^{2}}$ ,

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

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

$\Rightarrow \frac{d^{2} V}{d h^{2}} = 0 - \frac{3 π}{4} \times 2 h$

$\Rightarrow \frac{d^{2} V}{d h^{2}} = \frac{- 3 π h}{2}$

Since $\frac{d^{2} V}{d h^{2}} < 0$ for $h = \frac{2 R}{\sqrt{3}}$

$∴$ Volume is maximum for $h = \frac{2 r}{\sqrt{3}}$ .

Find the altitude of a right circular cylinder of maximum volume inscribed in a sphere of radius $r$ .

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Find the altitude of a right circular cylinder of maximum volume inscribed in a sphere of radius r.

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Find the altitude of a right circular cylinder of maximum volume inscribed in a sphere of radius $r$ .