Final answer:
To minimize the cost of constructing a rectangular box with a square base and top, you need to find the dimensions that minimize the cost.
Step-by-step explanation:
To minimize the cost of constructing a rectangular box with a square base and top, we need to find the dimensions that minimize the cost. Let's assume the length of the sides of the base and top are x, and the height of the box is h. The volume of the box is given as 735 ft³, which can be expressed as:
V = x² * h = 735
To find the dimensions that minimize the cost, we need to express the cost function in terms of x. The cost of the base is 15 cents per square foot, the cost of the top is 10 cents per square foot, and the cost of the sides is 1.5 cents per square foot. The cost function can be expressed as:
Cost = 2 * (15 * x²) + 1.5 * (4 * x * h)
To minimize the cost, we take the derivative of the cost function with respect to x and set it equal to zero:
dCost/dx = 2 * 15 * 2x + 1.5 * 4h = 0
Simplifying this equation, we get:
60x + 6h = 0
Now we can solve this system of equations to find the values of x and h that minimize the cost.