Optimization problem

Question

For an open-topped rectangular box, with a square base x by x cm and height h cm, find the dimensions giving the minimum surface area, given that the volume is 18 cm^3.

^{x = 3.301 cm}

what would be the height?

Answer 1

The area of a rectangle or square is L•W

So the area of the square base is x²

The area of the sides are x•h and there are 4 of those sides so 4xh

Therefore the total surface area is x² + 4xh

The volume of a rectangular box is L•W•H so the volume of this box is x²h therefore:

x²h = 18 meaning h = 18/x²

Plugging in "18/x²" into the surface area equation in place of "h" we get:

SA = x² + 4x(18/x²)

SA = x² + 72/x

SA(x) = x² + 72x^-1

To find the critical points, take the derivative and set it equal to zero:

SA'(x) = 2x - 72x^-2

2x - 72x^-2 = 0

2x = 72x^-2

2x³ = 72

x³ = 36

x = 3.302

To determine if this is a local min or a local max, look at the derivative for values of "x" below 3.302. Try x = 3

SA'(3) = 2(3) - 72(3^-2) = 6 - 8 = -2 meaning the function is decreasing.

Now look at the value of the derivative for x > 3.302, say x = 4:

SA'(4) = 2(4) - 72(4^-2) = 8 - 4.5 = 3.5 meaning the function is increasing.

Therefore x = 3.302 is a minimum.