Name: Rates of Change
Uploaded: 2019-07-19
Duration: 48 min 10 s
Description: In this video, let us know the average rate of change and instantaneous rate of change. Learn more about this concept in the video.

Question

Find the volume of the largest cylinder that can be inscribed in a sphere of radius '

r

'

cm

.

Accepted Answer

Given,

Let $R$ be the radius and $h$ be the height of the cylinder which is inscribed in a sphere of radius $r cm$ .

Then from the figure,

$R^{2} + {(\frac{h}{2})}^{2} = r^{2}$

$∴ R^{2} = r^{2} - \frac{h^{2}}{4} . . . . . . . . . . . (1)$

Let V be the volume of the cylinder.

Then $V = {πR}^{2} h$

$V = π (r^{2} - \frac{h^{2}}{4}) h$

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

Differentiating $(2)$ w.r.t. $h$ , we get

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

$\frac{d V}{d h} = π (r^{2} \times 1 - \frac{1}{4} \times 3 h^{2})$

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

Again differentiating $(3)$ w.r.t $h$ , we get

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

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

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

Now, $\frac{d V}{d h} = 0$ gives, $π (r^{2} - \frac{3}{4} h^{2}) = 0$

$∴ r^{2} - \frac{3}{4} h^{2} = 0$

$∴ r^{2} = \frac{3}{4} h^{2}$

$∴ \frac{4 r^{2}}{3} = h^{2}$

$∴ h = \sqrt{\frac{4 r^{2}}{3}} = \frac{2 r}{\sqrt{3}} . . . . . [∵ h > 0]$

From $(4)$ ,

${(\frac{d^{2} V}{d h^{2}})}_{a t h = \frac{2 r}{\sqrt{3}}} = \frac{- 3}{2} π \times \frac{2 r}{\sqrt{3}} < 0$

$∴ V$ is maximum at $h = \frac{2 r}{\sqrt{3}}$ .

If $h = \frac{2 r}{\sqrt{3}}$ , then from $(1)$

$R^{2} = r^{2} - \frac{1}{4} \times \frac{4 r^{2}}{3} = \frac{2 r^{2}}{3}$

$∴$ Volume of the largest cylinder $= π \times \frac{2 r^{2}}{3} \times \frac{2 r}{\sqrt{3}} = \frac{4 {πr}^{3}}{3 \sqrt{3}} cu cm .$

Hence, the volume of the largest cylinder inscribed in a sphere of radius $' r' cm = \frac{4 r^{3}}{3 \sqrt{3}} cu cm$ .

Find the volume of the largest cylinder that can be inscribed in a sphere of radius ' $r$ ' $cm$ .

Important Questions on Applications of Derivatives

Learn and Prepare for any exam you want !

Find the volume of the largest cylinder that can be inscribed in a sphere of radius 'r' cm.

Important Questions on Applications of Derivatives

Learn and Prepare for any exam you want !

Find the volume of the largest cylinder that can be inscribed in a sphere of radius ' $r$ ' $cm$ .