Name: Taylor and Maclaurin Series
Uploaded: 2019-07-19
Duration: 9 min 34 s
Description: Let's watch this video to learn about the Taylor series and a special case of those called the Maclaurin series. This uses differentiation and some polynomials

Question

A hollow cone with base radius

r

and height

h

is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is

\frac{4}{9}

times volume of the cone.

Accepted Answer

Given $r$ is the radius of hollow cone and $h$ is the height.

Let $x$ be the radius of the cylinder and $y$ be the distance from the top of the cone to the top of the inscribed cylinder.

Therefore, the height of the cylinder is $h - y$

The volume of the inscribed cylinder is $π x^{2} (h - y)$

By the method of similar ratios

$\frac{y}{x} = \frac{h}{r} \Rightarrow y = \frac{x h}{r}$

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

Now, $\frac{d V}{d x} = 2 π x h - 3 π x^{2} \frac{h}{r}$

For maximum value, equate the first derivative to zero.

$\Rightarrow 2 π x h - 3 π x^{2} \frac{h}{r} = 0 \Rightarrow x (2 - 3 x \frac{1}{r}) = 0 \Rightarrow x = 0, x = \frac{2 r}{3}$

At $x = 0, V = 0$

$\frac{d^{2} V}{d x^{2}} = 2 π h - 6 π x \frac{h}{r}$

$= 2 π h$ $> 0$

So $V$ is minimum at $x = 0$ .

At $x = \frac{2 r}{3}, V = \frac{4 π r^{2} h}{27}$

$\frac{d^{2} V}{d x^{2}}$ $= - 2 π h$ $< 0$

So $V$ is maximum at $x = \frac{2 r}{3}$ .

The maximum volume of cylinder $V = \frac{4 π r^{2} h}{27}$ .

$\Rightarrow V = \frac{4}{9} (π r^{2} \frac{h}{3})$

Hence, the volume of the largest cylinder that can be inscribed under cone is $\frac{4}{9}$ times the volume of the cone.

A hollow cone with base radius $r$ and height $h$ is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is $\frac{4}{9}$ times volume of the cone.

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A hollow cone with base radius r and height h is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is 49 times volume of the cone.

Important Questions on Applications of Differential Calculus

Learn and Prepare for any exam you want !

A hollow cone with base radius $r$ and height $h$ is placed on a table. Show that the volume of the largest cylinder that can be hidden underneath is $\frac{4}{9}$ times volume of the cone.