Applied minimum problems

Question

A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 500 cubic centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.04 cents per square centimeter. The top will be made of glued paper, costing 0.06 cents per square centimeter. Find the dimensions for the package that will minimize production cost.

Radius:

Height:

Minimum cost in cents:

Answer 1

Lily,

First we must find an equation to model the cost, the best way we can do this is to look at the areas of each part of the cylinder.

Let h = height of the cylinder

let r = radius of the cylinder

Surface area of the top/bottom: A = πr²

Surface area of the sides: A = 2πrh

So, let the cost be C

C = 2πrh(0.04) + πr²(0.04) + πr²(0.06)

This is restrained by the Volume: V = πr²h = 500cm³ ⇒ h = 500/πr²

Substitute this value for h into our cost equation to get:

C = πr[0.08(500/πr²) + 0.1r]

C = πr(40/πr² + 0.1r)

C = 40/r + 0.1πr²

⇒ C' = -40/r + 0.1πr²

Set C' = 0 and solve for r.

C' = -40/r + 0.1πr² = 0

⇒ r³ = 40/0.2π, so r ≈ 1.85 cm

h = 500/πr² so, h≈ 46.33 cm

∴ C = π(1.85)[0.08(46.33) + 0.1(1.85)] = 22.62 cents

To ensure we have a minimum we must find C"

C" = 80/r³ + 0.2π > 0 for all r >0, so our solution is indeed a minimum.

I hope this helps and if you need any clarification where work hasn't been shown, please feel free to ask!