A microwaveable cup-of-soup package needs to be constructed in the shape of cylinder to hold 350 cubic

centimeters of soup. The sides and bottom of the container will be made of styrofoam costing 0.03 cents per square

centimeter. The top will be made of glued paper, costing 0.08 cents per square centimeter. Find the dimensions for

the package that will minimize production cost.

Helpful information:

To minimize the cost of the package:

Radius:

Height:

Minimum

Question

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Respuesta :

jtellezd jtellezd · Answer 1 · 2019-12-01T23:01:38+01:00

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