Question

If a cylinder having maximum curved surface area, is inscribed in a sphere of radius 3 cm, then the volume (in cm3) of this cylinder is:

Answer 1

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Answer 2

Concept:

The volume of the cylinder is $\pi r^{2}h$.

Given:

Radius of Sphere= 3cm

Find:

Volume(V) of the cylinder.

Solution:

From the figure-

$(3)^2=(AC)^2+(h/2)^2$

∵ AC is the radius of cylinder so replace it with r

$r^2=9-h^2/4$

∵ V=$\pi r^{2}h$

$V=\pi (9-h^2/4)*h\\\\V=9\pi h-(\pi h^3)/4$

Since V is a function of h so on differentiating V w.r.t.h gives-

$dV/dh=9\pi -3/4 \pi h^2$

For Maxima/Minima $dV/dh=0$

$9\pi -3/4\pi h^2=0\\\\h^2=36\pi /3\pi \\\\h^2=12\\\\h=+\sqrt{12} ,-\sqrt{12}$

∵ $(d^2 V)/(d^2 h)=(-6) \pi h/4$

putting h= +$\sqrt{12}$ gives $(d^2 V)/(d^2 h) < 0$

hence V is max at h=+$\sqrt{12}$

$(d^2 V)/(d^2 h) < 0$ at h=-$\sqrt{12}$ hence it is minima

Now ∵$r^2=9-h^2/4$

$r=9-12/4\\\\r=3$

now for V

$V=\pi r^2 h$

r=3 and h=$\sqrt{12}$

$V=\pi *(3)^2*\sqrt{12}\\\\V=\pi *9*2\sqrt{3}\\\\v=18\pi \sqrt3$

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