Question

Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height h is $\frac{1}{3}$ h.

Solution

Let a cylinder be inscribed in a cone of radius R and height h.

Let a radius of the cylinder be r and its height be h₁.

It can be easily seen that $∆$ AGI and $∆$ ABD are similar.

$∴ \frac{AI}{AD} = \frac{GI}{BD} \Rightarrow \frac{h - h_{1}}{h} = \frac{r}{R} \Rightarrow r = \frac{R}{h} h - h_{1} Volume (V) of the cylinder = {πr}^{2} h_{1} \Rightarrow V = π \frac{R^{2}}{h^{2}} (h - h_{1})^{2} h_{1} \Rightarrow V = π \frac{R^{2}}{h^{2}} [h^{2} + {h_{1}}^{2} - 2 {hh}_{1}] x h_{1} \Rightarrow \frac{dv}{{dh}_{1}} = π \frac{R^{2}}{h^{2}} [h^{2} + {h_{1}}^{2} - 2 {hh}_{1} + h_{1} (2 h_{1} - 2 h)] \Rightarrow \frac{dv}{{dh}_{1}} = π \frac{R^{2}}{h^{2}} (h^{2} + 3 {h_{1}}^{2} - 4 {hh}_{1})$

$Now, \frac{dV}{{dh}_{1}} = 0 \Rightarrow \frac{{πR}^{2}}{h^{2}} (h^{2} + 3 {h_{1}}^{2} - 4 {hh}_{1}) = 0 \Rightarrow 3 {h_{1}}^{2} - 4 {hh}_{1} + h^{2} = 0 \Rightarrow 3 {h_{1}}^{2} - 3 {hh}_{1} - {hh}_{1} + h^{2} = 0 \Rightarrow 3 h_{1} (h_{1} - h) - h (h_{1} - h) = 0 \Rightarrow (h_{1} - h) (3 h_{1} - h) = 0 \Rightarrow h_{1} = h, h_{1} = \frac{h}{3}$

It can be noted that if h₁ = h, then the cylinder cannot be inscribed in the cone.

$∴ h_{1} = \frac{h}{3} Now, \frac{d^{2} V}{{dh}_{1}^{2}} = \frac{{πR}^{2}}{h^{2}} 0 + 6 h_{1} - 4 h = \frac{{πR}^{2}}{h^{2}} 6 h_{1} - 4 h ∴ {\frac{d^{2} V}{{dh}_{1}^{2}}}_{(h_{1} = \frac{h}{3})} = \frac{{πR}^{2}}{h^{2}} [\frac{6 h}{3} - 4 h] = \frac{- 2 {πR}^{2}}{h} < 0$

Therefore, by the second derivative test, h₁ = $\frac{h}{3}$ is the point of local maxima of V.

So, the volume of the cylinder is the maximum when h₁ = $\frac{h}{3}$ .

Hence, the height of the cylinder of the maximum volume that can be inscribed in a cone of height h is $\frac{1}{3} h$ .

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Application Of Derivatives

Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height h is $\frac{1}{3}$ h.

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For Daily Free Study Material Join wiredfaculty

Application Of Derivatives

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Some More Questions From Application of Derivatives Chapter

The radius of a circle is increasing uniformly at the rate of 4 cm per second Find the rate at which the area of the circle is increasing when the radius is 8 cm.

Mock Test Series

NCERT Sample Papers

Entrance Exams Preparation

Show that the height of the cylinder of maximum volume that can be inscribed in a cone of height h is $\frac{1}{3}$ h.