Question
Asked Oct 24, 2019

A cylinder is inscribed in a sphere with radius 5. Find the height h of the cylinder with the maximum possible volume.

check_circleExpert Solution
Step 1

Our aim is to inscribe a right circular cylinder inside a sphere with radius 5 cm such that the volume of the cylinder is maximum.

 

2
2
5-h
h
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2 2 5-h h

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Step 2

As it is clear from the figure above that,

radius of the cylinder = V52 - h2 = 25 - h2
height of the cylinder
2h
Thus, volume of the cylinder, V = nr2h
2
t(V25 - h2) (2h)
= T
2T(25 h2h
= 277(25h - h3)
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radius of the cylinder = V52 - h2 = 25 - h2 height of the cylinder 2h Thus, volume of the cylinder, V = nr2h 2 t(V25 - h2) (2h) = T 2T(25 h2h = 277(25h - h3)

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Step 3

We know th...

dV
=0
dh
For volume to be maximum,
dV
d
[2п(25h — h?)] — 0
=>
dh dh
2п (25 — Зh?) — 0
Зh? — 25
5
h =
У3
Л
Л
help_outline

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dV =0 dh For volume to be maximum, dV d [2п(25h — h?)] — 0 => dh dh 2п (25 — Зh?) — 0 Зh? — 25 5 h = У3 Л Л

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Tagged in

Math

Calculus

Derivative