A rectangular box with a square base and top is to be made to contain 1250 cubic feet. The material for the base costs 35 cents per square foot, for the top 15 cents per square foot, and for the sides 20 cents per square foot. Find the dimensions that will minimize the cost of the box.

Your top and bottom, averaged, are more expensive than the sides. Top and bottom: 15+35=50/2 = 25cents per square foot. Two sides: 20+20=40/2=20cents per square foot. therefore, you would want as little area on the bases as possible. If you place one square foot each on the top and bottom, that leaves 1248 square feet for the sides. 1248/4 = 312 square feet per side. The sides are one foot (length of base) x 312. Therefore, your box would be 312x1x1. This would be (1 sq ft x 15 cents) + (1 sq ft x 35 cents) + 1248 sq ft x 20 cents)

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To find the dimensions that will minimize the cost of the box, we need to come up with a cost function based on the given information.

Let's assume the base of the rectangular box has dimensions x by x, and the height of the box is h. Since the box has a square base, the area of the base is x² square feet.

The total volume of the box is given as 1250 cubic feet, so we have the equation:

x² * h = 1250

From this equation, we can express the height h in terms of x:

h = 1250 / x²

Now, let's find the cost function. The cost of the base will be x² multiplied by the cost per square foot for the base, which is 35 cents or $0.35:

Cost of the base = 0.35 * x²

Similarly, the cost of the top will also be x² multiplied by the cost per square foot for the top, which is 15 cents or $0.15:

Cost of the top = 0.15 * x²

The cost of the sides will be the sum of the cost per square foot for the sides times the total area of the sides. Since the box has 4 sides, the total area of the sides is 4 times the product of the base dimensions x by h:

Area of the sides = 4 * x * h

Cost of the sides = 0.20 * Area of the sides
= 0.20 * 4 * x * h
= 0.80 * x * h

To get the total cost, we add the costs of the base, top, and sides:

Cost = Cost of the base + Cost of the top + Cost of the sides
= 0.35 * x² + 0.15 * x² + 0.80 * x * h

Substitute the expression for h from the volume equation:

Cost = 0.35 * x² + 0.15 * x² + 0.80 * x * (1250 / x²)

Simplify the equation:

Cost = 0.50 * x² + 1000 / x

To minimize the cost, we need to find the critical points by taking the derivative of the cost function with respect to x:

dCost/dx = 1.00 * x - 1000 / x²

Set the derivative equal to zero and solve for x:

0 = 1.00 * x - 1000 / x²

Multiply both sides by x² to clear the equation of fractions:

0 = x³ - 1000

Solve for x:

x³ = 1000
x = ∛1000
x ≈ 10

Now that we have the value of x, we can substitute it back into the volume equation to find the corresponding height h:

h = 1250 / x²
h = 1250 / (10)²
h = 1250 / 100
h = 12.5

So, the dimensions that will minimize the cost of the box are approximately x = 10 and h = 12.5.