Name: Increasing Functions
Uploaded: 2019-07-19
Duration: 3 min 59 s
Description: The GDP of India is either increasing or decreasing, depending on a variety of factors. Let’s understand what an increasing function is, by using derivatives...

Question

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius

5 \sqrt{3} cm

is

(500 π) {cm}^{3}

.

Accepted Answer

Given, radius of sphere $= 5 \sqrt{3} cm$ , and let height of cylinder $= h cm$

To Prove: The maximum volume of the cylinder which can be inscribed in a sphere of given radius is $(500 π) {cm}^{3}$

Consider,

Let $R$ be the radius of the cylinder then from right-angled triangle $O A C$ , we have

$R^{2} = {(5 \sqrt{3})}^{2} - \frac{h^{2}}{4}$

$\Rightarrow R = \sqrt{75 - \frac{h^{2}}{4}}$

Now, we know that volume of cylinder is given by

$V = {πR}^{2} h$

$= π (75 - \frac{h^{2}}{4}) h$

$= π (75 h - \frac{h^{3}}{4}) . . . . (i)$

Now, $\frac{d V}{d h} = π (75 - \frac{1}{4} \times (3 h^{2})) . . . . (i i)$

For Maxima and minima points, we put

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

$\Rightarrow 300 - 3 h^{2} = 0$

$\Rightarrow h = 10 cm$

Now, $\frac{d^{2} V}{d h^{2}} = - π (\frac{1}{4} \times 3 \times 2 h)$ [Using $(i i)$ ]

$\frac{d^{2} V}{d h^{2}} |_{h = 10} = - π (\frac{1}{4} \times 3 \times 2 \times 10) < 0$

$\Rightarrow At h = 10 cm$ , volume of cylinder has a maximum value.

Now, from $(i)$ , we have

$V = π (75 \times 10 - \frac{{(10)}^{3}}{4})$

$= π (750 - 250)$

$= 500 π$ ${cm}^{3}$

Hence, Proved.

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5 \sqrt{3} cm$ is $(500 π) {cm}^{3}$ .

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius 53 cm is (500π)cm3.

Important Questions on Application of Derivatives

Learn and Prepare for any exam you want !

Show that the maximum volume of the cylinder which can be inscribed in a sphere of radius $5 \sqrt{3} cm$ is $(500 π) {cm}^{3}$ .