A box (with no top) is to be constructed from a piece of cardboard of sides A and B by cutting out squares of length h from the corners and folding up the sides. Find the value of h that maximizes the volume of the box if

A = 7 and B = 12

h = 1,743

Step-by-step explanation:

Volume of a box is

V (h) = (A - 2h) * (B - 2h) * h A = 7 B = 12

We have

V (h) = (7 - 2h) * (12 - 2h) * h

V (h) = (84 - 14*h - 24*h + 4*h²) * h

V (h) = (84 - 38*h + 4 * h²) * h ⇒ V (h) = 84h - 38h² + 4h³

Taking derivatives both sides of the equation

V' (h) = 84 - 76h + 12x²

V' (h) = 0 84 - 76h + 12x² = 0 42 - 38h + 6x²

3x² - 19h + 24 = 0

Solving for h h1 = [ (19 + √ (19) ² - 288 ] / 6 h1 = [ (19 + √73) / 6]

h₁ = 4,59 we dismiss this value since 9,18 (4,59*2) > A

h₂ = [ 19 - √73) / 6] h₂ = 1,743

h = 1.743 is h value to maximizes V