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 as in the figure below.

suppose that the box height is h=3 in. and that it is constructed using 137 in.2 of cardboard (i. e., ab=137). which values a and b maximize the volume?

We first identify the given values, AB = 137. With this information we can have an equation for any of the two unknown. Let's say A = 137/B.

The volume of the box would be V = LWH, where it becomes V = 3(A-6) (B-6) and we substitute the equation with our first one to make it 

V = 3(137/B-6)(B-6) and simplifying it we have, V= 519- 2466/B - 18B

Next we get its derivative so V' = -18 -2466/B^2 = 0 where we have B = √137 or 11.70. 

In getting A, we substitute it A = 137/11.70 = 11.70. The values of A and B are the same, √137 or 11.70 inches.

The density of the cube is 40