Math CentralQuandaries & Queries


Question from Haris, a student:

question: the cylinder below is to be made with 3000cm^2 of sheet metal. the aim of this assignment is to determine the dimensions (r and h) that would give the maximum volume.
how do i do this?
i have no idea. can you please send me a step-to-step guide on how t do this?
thank you very much.

Hi Haris,

The surface of the cylinder is composed of two circular disks of radius r cm each and the side of the cylinder which can be cut from bottom to top and rolled out to form a rectangle. (Think of the label on a soup can.) The rectangle has height h cm and its length is the circumference of the circular top which is 2 π r cm. The surface area of the cylinder is then 2 × π r2 + h × 2 π r cm2. You know the surface area is to be 3000 cm2 and hence

2 π r2 + 2 π r h = 3000 cm2                    (1)

The volume of a cylinder of radius r cm and height h cm is

V = π r2 h cm3                                       (2)

Solve equation (1) for h and substitute into equation (2). This will give an expression for the volume in terms of r alone. Use the calculus you know to maximize V.

I hope this helps,

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