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A right circular cylinder is inscribed in a sphere of radius a > 0. What is the height of the cylinder when its volume is maximal?

My attempt : i know that volume of cylinder = π(a^2)h

 Surface area = πah + π(a^2)h = S S = πah + π(a^2)h for Maximum , dV/dx = (1/2)(S- 6πa^2)= 0 S = 6πa^2 S = 6πa^2. i know that h = S -π(a^2)h/2πa i got h = 2a is the required answer

Is my answer is correct or not,,PLiz verified my solution

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1 Answer

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Let $\theta$ be the angle from the center of the sphere to the tip of the cylinder. Then

$$ r=a\cos\theta\\ h=2a\sin\theta\\ V=\frac{\pi rh}{3}=\frac{2\pi a^3}{3}\cos^2\theta\sin\theta $$

Setting $dV/d\theta=0$ to find the maximum value, we get

$$-2\cos\theta\sin^2\theta+\cos^3\theta=-2\tan^2\theta+1=0\\ \theta=\tan^{-1}\frac{1}{\sqrt{2}}\approx 35.262^{\circ} $$

Then $h$ and $r$ are given as above. If it's of interest, the surface are of the cylinder is given by (not including the top and bottom) is given by

$$S=2\pi r\sqrt{r^2+h^2}=2\pi a^2\sqrt{\cos^2\theta+2\sin^2\theta}=2\pi a^2\sqrt{1+\sin^2\theta}$$

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