Finding the Maximum dimension of a box.

Question

You want to make an open box having a square base and a surface area of 108 square inches. What dimensions will produce a box with a maximum volume?

Answer 1

Let length = width = x

Let height = h

 

Using the formula for surface area,

 

x² + 4xh = 108

 

 

Using the formula for volume,

 

V = x²h

 

 

We can substitute the surface area equation into the volume equation.

 

From SA,

 

h = (108 - x²) / 4x

h = 27x^-1 - (1/4)x

 

Substituting,

 

V = x²(27x^-1 - (1/4)x)

V = 27x - (1/4)x³

 

 

Now, we set the derivative of V equal to zero.

 

V' = 0

 

27 - (3/4)x² = 0

 

     -(3/4)x² = -27

 

             x² = -27(-4/3)

             x² = 36

             x = -6     or      x = 6

 

 

The length and width are 6 in

 

Now we find the height.

 

h = (108 - 6²) / (4 * 6)

h = (108 - 36) / 24

h = 72 / 24

h = 3

 

The height is 3 in.