How to solve such an optimization problem efficiently？?

How to solve this optimization problem?

• The volume of a square based rectangular cardboard box needs to be 1000 cm^3. Determine the dimensions that require the minimum amount of material to manufacture all six faces. Assume that there will be no waste material. The machinery available cannot fabricate material smaller than 2cm in length

• Answer:

  Suppose that the dimensions of the box are x by x by h (note that the base of the box is a square, so the length and width are the same). The volume of the box will then be: V = LWH = (x)(x)(h) = x^2*h, and its surface will be: S = 2LW + 2LH + 2WH = 2(x)(x) + 2(x)(h) + 2(x)(h) = 2x^2 + 4xh. Since the volume of the box is 1000 cm^2, we see that: x^2*h = 1000. In order to minimize S, we need to express S in terms of one variable. Right now, S is in terms of x and h; however, we can use the fact that x^2*h = 1000 to express h in terms of x and, thus, represent S in terms of x. Solving x^2*h = 1000 for h gives: h = 1000/x^2. So, the surface area of the box in terms of x is: S = 2x^2 + 4x(1000/x^2) = 2x^2 + 4000/x. To minimize S, we need to take its derivative and set it equal to zero. This gives: dS/dx = 4x - 4000/x^2 = 0. Multiplying both sides by x^2 gives: 4x^3 - 4000 = 0 ==> x^3 = 1000 ==> x = 10. Using the second derivative test, x = 10 is a minimum of S; in fact, we can argue that due to the geometrical nature of this problem, x = 10 is a global minimum of S. Then, since x = 10, we see that h = 1000/x^2 gives h to be: h = 1000/10^2 = 10. Therefore, the required dimensions of the box are 10 cm by 10 cm by 10 cm. I hope this helps!