A closed right circular cylinder must enclose a fixed volume

MATHEMATICS · Single correct · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

V

. Its surface area is

S(r,h)=2\pi r h+2\pi r^{2}

. Which statement best explains why it is valid to replace

h

with

V/(\pi r^{2})

before determining the value of

r

that minimizes

S

? (a) Because

\dfrac{\partial S}{\partial h}=2\pi r

, which is never zero, the optimum forces

h

to be fixed by the volume constraint; substituting

h=V/(\pi r^{2})

lets us minimise only with respect to

r

. (b) The constraint

V=\pi r^{2}h

uniquely relates

h

r

, so expressing

h

in terms of

r

reduces

S

to a single-variable function whose extrema determine the minimum. (c) Since the cylinder with least surface area must use the smallest possible height, replacing

h

V/(\pi r^{2})

guarantees the height is minimised while the volume remains fixed. (d) Because most of the surface area comes from the curved side

2\pi rh

, treating

h

as a constant obtained from

h=V/(\pi r^{2})

provides a reliable approximation to the minimum.

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