Answer:
x = r = 3,12 cm
h = 11,45 cm
C(min) = 10,09 $
Step-by-step explanation:
Let x be radius of base and h height of cylinder
Let A₁ be Cylinder area of base + lateral area
A₁ = π*x² + 2*π*x*h Now V = 350 cm³ 350 = π*x²*h
h = 350/ π*x²
Then
A₁ = π*x² + 700/x
A₂ area of the top
A₂ = π*x²
Now we write the expression of cost as fuction of x
C(x) = 0,03 * ( π*x² + 700/x) + 0,08*π*x²
C(x) = 0,03* π*x² + 21/x + 0,08*π*x²
C(x) = 0,11* π*x² + 21/x
Taking derivatives on both sides of the equation
C´(x) = 2*0,11*π*x - 21 / x²
C´(x) = 0 0,22*π*x - 21 / x² = 0
0,22*π*x³ - 21 = 0 0.6908* x³ - 21 = 0
X³ = 21/0,6908 X³ = 30,40
x = 3,12 cm
And
h = 350/π*x² h = 350 / 30,57
h = 11,45 cm
C(min) = 0,11* π*(3,12)² + 21/3,12
C(min) = 3,36 + 6,73
always true
so 
is never true