The radius of the cylinder of maximum volume, | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(B) Let $r$ be the radius and $h$ be the height of the cylinder inscribed in a sphere of radius $R$. From the geometry of the figure,we have the relat

Vedclass · Accepted Answer

$\sqrt {{2 \over 3}} R$

Vedclass · Answer

(B) Let $r$ be the radius and $h$ be the height of the cylinder inscribed in a sphere of radius $R$. From the geometry of the figure,we have the relation:${r^2} + {\left( {\frac{h}{2}} \right)^2} = {R^2}$${h^2} = 4({R^2} - {r^2}) \implies h = 2\sqrt {{R^2} - {r^2}}$The volume $V$ of the cylinder is given by:$V = \pi {r^2}h = 2\pi {r^2}\sqrt {{R^2} - {r^2}}$To find the maximum volume,we differentiate $V$ with respect to $r$:$\frac{{dV}}{{dr}} = 2\pi \left[ {2r\sqrt {{R^2} - {r^2}} + {r^2} \cdot \frac{1}{{2\sqrt {{R^2} - {r^2}} }} \cdot ( - 2r)} \right]$$\frac{{dV}}{{dr}} = 2\pi \left[ {2r\sqrt {{R^2} - {r^2}} - \frac{{{r^3}}}{{\sqrt {{R^2} - {r^2}} }}} \right]$Setting $\frac{{dV}}{{dr}} = 0$ for critical points:$2r\sqrt {{R^2} - {r^2}} = \frac{{{r^3}}}{{\sqrt {{R^2} - {r^2}} }}$$2({R^2} - {r^2}) = {r^2}$$2{R^2} - 2{r^2} = {r^2}$$3{r^2} = 2{R^2}$${r^2} = \frac{2}{3}{R^2}$$r = \sqrt {\frac{2}{3}} R$Thus,the radius of the cylinder of maximum volume is $\sqrt {\frac{2}{3}} R$.

The radius of the cylinder of maximum volume,which can be inscribed in a sphere of radius $R$ is

Similar Questions

If $a_n$ is the greatest term in the sequence $a_n = \frac{n^3}{n^4+147}$,$n = 1, 2, 3, \ldots$,then $n$ is equal to $..........$.

The function $f(x) = ax + \frac{b}{x}$ where $a, b, x > 0$ takes on the least value at $x$ equal to:

If $f(x) = \sqrt{x^2 + x} + \frac{\tan^2 \alpha}{\sqrt{x^2 + x}}$,where $\alpha \in (0, \pi/2)$ and $x > 0$,find the minimum value of $f(x)$.

The maximum area of a rectangle that can be inscribed in a circle of radius $2 \text{ units}$ is

The sum of two numbers is $20$. If the product of the square of one number and the cube of the other is maximum,then the numbers are:

Vedclass Test Series

Exam Paper Generator

Online Exam Module