Question

A box with a square base and open top just have volume of 32,000 cm^3 . find the dimension of the box that minimizes the amount of material used.

Answer 1

The volume of a box with a square base and open top:

V = Ab ∙ h

Ab = Area of base = x²

V = Ab ∙ h = x² ∙ h

x² ∙ h = 32,000

Divide both sides by x²

h = 32,000 / x²

The surface area of the box = Area of base + 4 Area of rectangle

As = x² + 4 Ar

Ar = Area of rectangle = x ∙ h

As = x² + 4 ∙ x ∙ h

As = x² + 4 ∙ x ∙ 32,000 / x²

dAs / dx = As'₍ x₎ = ( x² + 4 ∙ x ∙ 32,000 / x² )' = 2 ( x³ - 64,000 ) / x²

If

f ' (x) = 0

function has a local maximum or a local minimum

As'₍ x₎ = 0

2 ( x³ - 64,000 ) / x² = 0

Multiply both sides by x² / 2

x³ - 64,000 = 0

Add 64,000 to both sides

x³ = 64,000

x = ∛64,000 = 40

x = 40 cm

The second derivative test:

When

f ' ₍x₀₎ = 0

then

if

f " ₍x₀₎ > 0, then f has a local minimum at x₀

if

f " ₍x₀₎ < 0, then f has a local maximum at x₀

In this case:

As"₍ x₎ = ( As'₍ x₎ )' = [ 2 ( x³ - 64,000 ) / x² ] ' = 256,000 / x³ + 2

for x = 40

As" = 256,000 / x³ + 2 = 256,000 / 40³ + 2 = 6 > 0

So the surface area of the box has a local minimum for x = 40 cm

Dimension of the box that minimizes the amount of material used:

x = 40 cm

h = 32,000 / x² = 32,000 / 40² = 20 cm