Final answer:
To minimize the surface area of an open box with a square base and volume of 2000 m³, we find the dimensions that maximize the volume. The dimensions that minimize the surface area are a square base with side length of 20 m and a height of 5 m.
Step-by-step explanation:
To minimize the surface area of the open box, we need to find the dimensions that maximize the volume. Let's assume that the side length of the square base is x and the height of the box is h. The volume of the box is given by V = x^2 * h. We are given that the volume needs to be 2000 m³, so we can write the equation x^2 * h = 2000.
To minimize the surface area, we need to find the sides of the square base that minimize the perimeter, which is 4x. Since x^2 * h = 2000, we can solve for h in terms of x: h = 2000 / x^2. Substituting this into the perimeter equation, we have P = 4x + 2x * (2000 / x^2) = 4x + 4000 / x.
To minimize the surface area, we need to minimize the perimeter, so we can take the derivative of P with respect to x and set it equal to zero to find the critical points. Differentiating P with respect to x, we get dP/dx = 4 - 4000 / x^2. Setting this equal to zero and solving for x, we find x = 20. Substituting this back into the volume equation, we can solve for h: h = 2000 / (20^2) = 5 m. Therefore, the dimensions of the box that minimize the surface area are a square base with side length of 20 m and a height of 5 m.