Question

A storage box with a square base must have a volume of 90 cubic centimeters. The top and bottom cost $0.20 per square centimeter and the sides cost $0.10 per square centimeter. Find the dimensions that will minimize cost. (Let x represent the length of the sides of the square base and let y represent the height. Round your answers to two decimal places.)

Answer 1

Answer:

x = 2,24 cm

y = 17,96 cm

Step-by-step explanation:

The volume of a cube is:

V = x²*y      where  x is the side of the square base and the height of the cube

V = 90 cm³

And the surface area of the cube is:

S(c)  =  Area of the base (A₁ ) + Lateral area (A₂)

A₁  = x²       and  V = x²*y     then  y = 90/x²

Therefore total cost is:

Cost of the top plus bottom  C₁ =  0,20 * 2 * x²  C₁ = 0,4*x²

Lateral cost (C₂)  is    C₂ = 0,10 * x*y    then  C₂ =  0,10*x ( 90/x²)    C₂ = 9/x

C(x) = 0,4*x² + 9/x

We take derivatives on both sides of the equation to get:

C´(x) = 0,8*x  -  9/x²

C´(x) =  0     0,8*x  - 9/x²  = 0

0,8*x³ - 9  = 0

0,8 * x³  = 9

x³  =  11,25

x = ∛ 11,25

x  =  2,24  cm

And   y  =  90 / 5,01        y  =  17,96 cm