Answer:
C(min) = 0.5*V + √V/1.256 $
Step-by-step explanation:
The volume of a circular cylinder is: V(c) = π*r²*h where r is the radius of the circumference of the base and h is the height
The cost of the can is = the cost of (base and top) + lateral cost
Base surface = top surface = π*r²
Then cost of ( base + top ) is = (2* π*r² )*0,1
Lateral surface is = 2*π*r*h
Then cost of lateral surface is: (2*π*r*h)*0,5
Total cost C(t) = (2* π*r² )*0,1 + (2*π*r*h)*0,5
V = π*r²*h
Total cost as a function of (V >0 a parameter) and r then
h = V / π*r²
C(V,r) = (2* π*r² )*0,1 + π*r*(V / π*r²)
C(V,r) = 0.2*π*r² + V*/r
Taking derivatives on both sides of the equation we get:
C´(V,r) = 2*0.2*π*r - V/r²
C´(V,r) = 0 0.4*π*r - V/r = 0
Solving for r
0.4*π*r² - V = 0 ⇒ 1.256*r² = V r = √ V/ 1.256 cm
and h = V /π * (√ V/ 1.256)²
h = 1/ 1.256*π
h = 0.254 cm
C(V,r) = 0.2*π*r² + V*/r
C(min) = 0.2*π* (√ V/ 1.256)² + V/ √ V/ 1.256
C(min) = 0.2*π*V/1.256 + V/ √ V/ 1.256
C(min) = 0.5*V + √V/1.256 $
