Final answer:
To minimize packaging costs, we need to find the dimensions of the box that can be constructed at minimum cost. The dimensions of the box are x, x, and y, and we can express the total cost as C = 0.23x² + 0.24xy. By taking the partial derivatives of C and setting them equal to zero, we find x = 1 ft and y = 6 ft.
Step-by-step explanation:
To minimize the packaging costs of the rectangular box, we need to find the dimensions of the box that can be constructed at minimum cost. Let's assume the length of the base of the box is x and the height of the box is y. Since the box is rectangular with a square base, the dimensions of the box are x, x, and y. We also know that the volume of the box is 12 ft³, so we have the equation x²y = 12.
The cost of the base material is $0.11/ft², the cost of the side material is $0.06/ft², and the cost of the top material is $0.07/ft². The total cost of the box is the product of the cost per unit area and the total area of the box. We can express the total cost as C = 0.11x² + 4(0.06xy) + 0.07x² = 0.23x² + 0.24xy.
To find the dimensions that minimize the cost, we can minimize the cost function C with respect to x and y. Taking the partial derivatives of C with respect to x and y, we get ∂C/∂x = 0.46x + 0.24y and ∂C/∂y = 0.24x. Setting these derivatives equal to zero and solving the resulting system of equations, we find x = 1 ft and y = 6 ft.