Final answer:
To find the box dimensions that minimize cardboard use for a given volume of 4,000 cm3, set up an equation for surface area in terms of length and width, apply calculus to optimize the function, and solve for the critical points.
Step-by-step explanation:
To minimize the amount of cardboard used for a box with a volume of 4,000 cm3 (4,000 ml), we need to find the dimensions that result in the smallest surface area. The volume of the box can be expressed as V = xyz, where x, y, and z are the length, width, and height, respectively. Since we want to find the smallest surface area, we can define the surface area function as S = xy + 2xz + 2yz. To find the minimum surface area, we need to use calculus. First, we express one variable in terms of the others using the volume constraint 4,000 = xyz. We could, for example, express z as z = 4,000 / (xy) and substitute this into the surface area function to get a function of two variables. We then find the partial derivatives with respect to x and y, set them to zero, and solve the system of equations to find the critical points. Analyzing these points will give the dimensions that minimize the amount of cardboard used.