# 02 - Cylinder of maximum convex area inscribed in a sphere

**Problem 02**

A cylinder is inscribed in a given sphere. Find the shape of the cylinder if its convex surface area is a maximum.

**Solution 02**

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Convex surface area of cylinder,

$A = \pi dh$

$A = \pi dh$

Where:

$d = D \cos \theta$

$h = D \sin \theta$

$A = \pi (D \cos \theta)(D \sin \theta)$

$A = D^2 \pi \cos \theta \sin \theta$

$\dfrac{dA}{d\theta} = D^2\pi(\cos^2 \theta - \sin^2 \theta) = 0$

$\sin^2 \theta = \cos^2 \theta$

$\tan^2 \theta = 1$

$\theta = 45^\circ$

$d = D \cos 45^\circ = 0.707D$

$h = D \sin 45^\circ = 0.707D$

$\text{diameter} = \text{height}$ *answer*