Problem

Show that height of the cylinder of greatest volu…

Problem 17 Easy Difficulty

Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius $\mathrm{R}$ is $\frac{2 \mathrm{R}}{\sqrt{3}}$. Also find the maximum volume.

Answer

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Calculus 1 / AB

NCERT Class 12 Part 1 - Math

Chapter 6

Application of Derivatives

Section 6

Maxima and Minima

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Video Transcript

in this question we have to show that the height of the cylinder of maximum volume that can be inscribed in its fell of the result is to arbitrary good trade like we have to find the maximum volume also. So let's there guys fed and a cylinder is inscribed inside there is Fair lady. Okay late. So in this triangle let's suppose expected areas of the cylinder. Yeah edged by the height of the cylinder and are better the areas of that is fair. Mhm. Right. So in this figure in this triangle we can I play the fight a good student. So the base would be the areas of the cylinder. That is X squared height would be etched by two. So it's by two square. And um high potency is the ideas of this fair. That is our square. Right So we can write access. Square is a close to our square minus as you square by 4th lady. Yeah. Therefore the volume of the cylinder. Mhm. We would be equal to buy access. Well edge. Right so the volume would be equal to pi We can substitute the value of X square. That is all square minus as you square by fourth. Multiplied by edge. So the volume is by us by all the scrimmage Minor Search Cube by 4th. Yeah, they so for the maximum volume we have to differentiate The volume with respect to hide and we are very quiet. This 20. So the differentiation would be ah the square minus 3" square by four. This should be equal to jesus. So from here we get the height of the cylinders for Albania. Italy four hours square by three square dude, which is equals two. To buy this credit after the right. So the height of the season, there should be two are by another three. Now we can find the volume of the cylinder. Volume would be equal to um by access quickly by access coverage. Right before that um we can find the value of the or we can find the volume of the cylinder film this equation made we can substitute the value of height here so we get, the volume of the cylinder is by multiplied by oh that's good to al by three under three minus one by four. four hours square battery Multiplied by two hours by Under three. So we have substituted the value of the height so we get the volume of the cylinder. As when we simplify this, the volume is fall by Fall by our Cube divided by three square root of the, so this is the volume of the cylinder.

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