Question

Find the volume of the largest cylinder that can be inscribed in a sphere of radius r centimetre

Answer 1

Step 1:

Let hh be the height and RR be the radius of the base of the inscribed cylinder.

Let VV be the volume of the cylinder

V=πR2hV=πR2h

By applying pythagoras theorem we get,

r2=(h2)2r2=(h2)2+R2+R2

Where rr is the radius of the sphere

∴R2=r2−(h24)∴R2=r2−(h24)

Substituting this in volume VV we get,

V=π(r2−h24)V=π(r2−h24)hh

⇒V=πr2h−π4⇒V=πr2h−π4h3h3

Step 2:

Now differentiating w.r.t hh we get,

dVdhdVdh=πr2−3πh24=πr2−3πh24

Again differentiating w.r.t hh we get,

d2Vdh2d2Vdh2=−3πh2=−3πh2

Since d2Vdh2d2Vdh2<0<0,V is maximium.

Step 3:

For maximum values of VV,we have

dVdhdVdh=0=0

⇒πr2−3πh24⇒πr2−3πh24=0=0

⇒πr2=3πh24⇒πr2=3πh24

⇒r2=3h24⇒r2=3h24

Step 4:

Take square root on both sides

r=3–√h2r=3h2

∴h=23–√∴h=23rr

Hence VV is maximum when h=23–√h=23rr

Put h=2r3–√h=2r3 in R2=r2−h24R2=r2−h24

Step 5:

We obtain R=23−−√rR=23r

The maximum volume of the cylinder is given by

V=πR2hV=πR2h

=π(23=π(23r2)(2r3–√)r2)(2r3)

=4πr^333/3√3

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