An open top box with a square base is to be made so that it holds 3 cubic feet. Assuming the material on the base costs $3 per square foot and the material on the sides costs $2 per square foot, determine the size of the base that minimizes the total cost.

let be base be x ft by x ft

let the height be y ft

(x^2)(y) = 3
y = 3/x^2

Cost = 3(x^2) + 2(4xy)
= 3x^2 + 8x(3/x^2)
= 3x^2 + 24/x

d(cost)/dx = 6x - 24/x^2
= 0 for a min of cost

6x = 24/x^2
x^3 = 4
x = cuberoot(4)
= appr 1.6 ft

the base should be appr 1.6 ft by 1.6 ft

(the height is appr 1.19 ft)

To solve this problem, we need to determine the dimensions of the square base that will minimize the total cost of building the box.

Let's start by assigning variables:
Let x be the length of one side of the square base.
Let y be the height of the box.

The volume of the box is given as 3 cubic feet, so we have the equation:
x * x * y = 3.

The surface area of the base will be x * x = x^2 square feet, and the surface area of the four sides will be 4 * x * y = 4xy square feet.

The total cost of materials is the sum of the cost of the base and sides. The cost of the base is 3 dollars per square foot, so it will be 3 * x^2 dollars. The cost of the sides is 2 dollars per square foot, so it will be 2 * 4xy = 8xy dollars.

To find the total cost, we can write the equation:
Total Cost = Cost of Base + Cost of Sides = 3 * x^2 + 8xy.

Now, we can express the equation for the volume (x * x * y = 3) in terms of y:
y = 3 / (x * x).

Substitute this value of y into the equation for the total cost:
Total Cost = 3 * x^2 + 8x * (3 / (x * x)).

Simplify the equation:
Total Cost = 3 * x^2 + 24 / x.

To minimize the total cost, we need to find the value of x that minimizes this equation.

We can do this by taking the derivative of the equation with respect to x, setting it to zero, and then solving for x.

d(Total Cost) / dx = 6x - 24 / x^2 = 0.

Multiply both sides by x^2 to get rid of the denominator:
6x^3 - 24 = 0.

Solve for x:
6x^3 = 24,
x^3 = 4,
x = ∛4.

So, the size of the base that minimizes the total cost is x = ∛4.

To calculate the corresponding height of the box (y), substitute the value of x into the equation for y:
y = 3 / (x * x) = 3 / (∛4 * ∛4) = 3 / (∛16) = 3 / 2 = 1.5.