Question

A cylindrical container that has a capacity of 4000 cubic centimeters is to be produced. The top and bottom of the container are to be made of material that costs $0.50 per square centimeter, while the sides of the container are to be made of material costing $0.40 per square centimeter. Find the dimensions that will minimize the total cost of the container.

Answer 1

the optimal size of the cylinder to minimise cost is

radius= R = 7.985 cm ≈ 8 cm

height =h = 19.969 cm ≈ 20 cm

Explanation:

Since the cost function is

C= 2*π*R²*c₁+ 2*π*R*h*c₂

where R = cylinder radius , h = height ,c₁= cost of the top material ,c₂ = cost f the side material

The volume of the cylinder is:

V=π*h*R² → h= V/(π* R²)

then

C= 2*π*R²*c₁ + 2*π*R* V/(π* R²) *c₂

C= 2*π*R²*c₁ + 2*π*c₂* V/(π* R) *c₂

the value of h that minimises the cost can be found through dC/dR=0 . Thus

dC/dR= 4*π*R*c₁ - 2*π*c₂* V/(π* R²) = 0

2*R*c₁ - c₂* V/(π* R²) = 0

2*R³*c₁ = c₂* V/π

R = ∛[(c₂/c₁)* (V/(2*π)]

replacing values

R = ∛[(c₂/c₁)* (V/(2*π)] = ∛ [($0.40/$0.50)*4000 cm³/(2*π)] = 7.985 cm

R = 7.985 cm

and the height would be

h= V/(π* R²) = 4000 cm³/[π*(7.985 cm)²]] = 19.969 cm

h= 19.969 cm