Calculus II Math 200 UBC Help :( Please leave a detailed solution as I am trying

Question

Calculus II Math 200 UBC Help :( Please leave a detailed solution as I am trying to understand how this was solved.

3. Cylindrical containers are made from sheets of metal from which the manufacturer cuts and bends the cylindrical surface plus its two lids which are the base discs of the cylinder. In order to minimize the cost of producing the containers, each holds 2000 units of volume, the manufacturer wants to design such a container with the minimal surface area. Find the dimensions of the con- tainer.

Explanation / Answer

volume = r2h = 2000

==> r2h = 2000

==> h = 2000/r2

surface area S = 2rh + r2

==> S(r) = 2r(2000/r2) + r2 = 4000/r + r2

minimum cost ==> S '(r) = 0

==> -4000/r2 + 2r = 0

==> r = 2000/r2

==> r3 = 2000

==> r = 12.599

==> h = 2000/r2 = 2000/12.5992

==> h = 12.599

Hence radius = 12.599 units , height = 12.599 units

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