The maximum volume (in cubic units) of the cy | vedclass

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(A) Let $h$ be the height and $r$ be the radius of the cylinder inscribed in a sphere of radius $R = 3$ units.From the geometry of the inscribed cylin

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$12 \sqrt{3} \pi$

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(A) Let $h$ be the height and $r$ be the radius of the cylinder inscribed in a sphere of radius $R = 3$ units.From the geometry of the inscribed cylinder,we have the relation $r^2 + (h/2)^2 = R^2$.Substituting $R = 3$,we get $r^2 + h^2/4 = 9$,which implies $r^2 = 9 - h^2/4$.The volume $V$ of the cylinder is given by $V = \pi r^2 h$.Substituting $r^2$,we get $V = \pi (9 - h^2/4) h = \pi (9h - h^3/4)$.To find the maximum volume,we differentiate $V$ with respect to $h$:$\frac{dV}{dh} = \pi (9 - 3h^2/4)$.Setting $\frac{dV}{dh} = 0$,we get $9 - 3h^2/4 = 0$,which implies $3h^2/4 = 9$,so $h^2 = 12$ and $h = 2\sqrt{3}$ (since height must be positive).Then $r^2 = 9 - (12/4) = 9 - 3 = 6$.The maximum volume is $V = \pi (6) (2\sqrt{3}) = 12\sqrt{3}\pi$ cubic units.

The maximum volume (in cubic units) of the cylinder which can be inscribed in a sphere of diameter $6$ units is

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