Find the volume of the largest right circular cylinder that can inscribed in a sphere of radius r cm - Maths - Application of Derivatives

Question

Find the volume of the largest right circular cylinder that can
inscribed in a sphere of radius r cm

ramandeep.singh · Accepted Answer

Let h be the height and R be the radius of base of the cylinder. Let V be the volume of the cylinder.

$∴ V = π R^{2} h ....... (1)$

In ΔOCA, we have

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

For maximum or minimum value of V,

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

Now

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

Thus, V is maximum when $h = \frac{2}{\sqrt{3}} r$ .

Therefore , R is calculated as

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

Thus, maximum volume of the cylinder is given by

$V_{\max} = π R^{2} h = π (\frac{2}{3} r^{2}) (\frac{2}{\sqrt{3}} r) = \frac{4 π r^{3}}{3 \sqrt{3}}$

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