Hello,

As per the question, we have to inscribe a cylinder into a sphere. Radis of sphere is = 12 cm

Let r is the radius and h is the height of this cylinder.

So when we make figure of this question, we found an equation i.e. (h/2)^2 + r^2 = R^2

Solving it we get , h = 2 ( √ R^2 - r^2 )

Now volune of cylinder V is = pi . r^2 . h

Put value of this h into this equation, we get

V = 2. pi. r^2. ( √ R^2 - r^2 )

We have to maximize this volume, so do dV/dr = 0

Solving it we get r^2 = 2. R^2/3

put R = 12 here, so r = 4√6

and h = 8√3

Hope it helps.