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.

the optimal size of the cylinder to minimise cost is

radius = R = 7.985 cm ≈ 8 cm

height = h = 19.969 cm ≈ 20 cm

Step-by-step 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