Abox (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