Optimization Problems

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 = 8 and B = 5.

Answer 1

V=(8-2h)(5-2h)h

= 4h^3 -26h^2 +40h

take the derivative, set=0

V'=12h^2 -52h +40= 0

divide by 4

3h^2 -13h +10=0

factor

(h-1)(3h-10)=0

set factor =0

h-1=0

h=1

V= (8-2)(5-2)=18 = max volume

ignore 3h-10=0, h=10/3, as it makes V<0