Answer:
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
