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 Incoming links: Maximal cylinder and box in an elliptic paraboloid

Past exam of the mathematics course of the year 1 of the natural sciences course of the University of Cambridge / 2017 / Paper 2 / 19 / b / Solution 2026-10-18

Use the diagram's labels:

a, b

are horizontal side lengths and

c

is the height of the rectangular parallelepiped. Its four top vertices are at horizontal distance

a^{2} + b^{2} /2

from the axis. Thus the contact constraint and volume are

c + \frac{H ( a ^{2} + b ^{2} )}{4 R ^{2}} = H, V_{p} = ab c .

Apply a Lagrange multiplier:

L = ab c - λ (c + \frac{H ( a ^{2} + b ^{2} )}{4 R ^{2}} - H) .

For positive dimensions, the derivative equations are

b c = λ \frac{H a}{2 R ^{2}}, a c = λ \frac{H b}{2 R ^{2}}, ab = λ .

Substituting the last equation into the first two gives

c = \frac{H a ^{2}}{2 R ^{2}} = \frac{H b ^{2}}{2 R ^{2}} .

Hence

a = b

, and the constraint becomes

2 c = H

. Therefore

a = b = R, c = \frac{H}{2}, V_{p, m a x} = \frac{R ^{2} H}{2} .

These are the maximal cylinder and box in an elliptic paraboloid dimensions for the rectangular case. A horizontal rotation of the square footprint gives the same volume by rotational symmetry.

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