Question

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

Answer 1

h = 1,743

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