17. Show that the height of the cylinder of maximum volume that can be inscribed in
a sphere of radius R is 2R
3
. Also, find the maximum volume.
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Answers
Step-by-step explanation:
Radius of the sphere =R
Let h be the diameter of the base of the inscribed cylinder.
Then h
2
+x
2
=(2R)
2
h
2
+x
2
=4R
2
.....(1)
Volume of the cylinder =πr
2
h
V=π(
2
x
2
)
2
.h
=π
4
x
4
.h
Volume =
4
1
πx
2
h
Substituting the value of x
2
, we get
V=
4
1
πh(4r
2
−h
2
)
From (1), we have
x
2
=4R
2
−h
2
V=πR
2
h−
4
1
πh
3
Differentiating with respect to x,
V=πr
2
h−
4
1
πh
3
dh
dV
=πR
2
−
4
3
πh
3
=π[R
2
−
4
3
h
2
]
We know
h
dV
=0
π[R
2
−
4
3
h
2
]=0
R
2
=
4
3
h
2
h=
3
2R
Also
dh
2
d
2
V
=−
4
3
.2πrh
=−
2
3
πh
At h=
3
2R
dh
2
d
2
V
=−
2
3
π(
3
2R
)=−ve
⇒V is maximum at h=
3
2R
Maximum volume at h=
3
2R
=
4
1
π[
3
2R
][4R
2
−
3
4R
2
]
=
2
3
πR
[
3
8R
2
]
=
3
3
4πR
3
sq. units
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