A packaging company is going to make open-topped boxes, with square bases, that hold 140 cubic centimeters. What are the dimensions of the box that can be built with the least material? (Round your answers to the nearest hundredth.)

x = 6.54 cm

h = 3.27 cm

Step-by-step explanation:

Volume of open top box

V = 140 cm³

Dimensions of the box

It is a base square box then area of the base of side x is

A (b) = x²

And we will call h the height of the box then

V = 140 ⇒ 140 = x²*h ⇒ h = 140 / x²

We have to calculate the area of the 4 sides

Area of one side is As = x*h ⇒ total area of 4 sides = 4 x * 140/x²

Ats = 560/x

Then Total area of the box is

A (t) = Area of the base + Total area of sides

A (x) = x² + 560/x

Taking derivatives on both sides of the equation we get:

A' (x) = 2x - 560/x²

A' (x) = 0 ⇒ 2x - 560/x² = 0 ⇒ 2x³ - 560 = 0

x³ = 280 ⇒ x = 6.54 cm

And h h = 140 / (6.54) ² ⇒ h = 140 / 42.77 h = 3.27 cm