Ignoring the walls’ thickness, determine the outside dimensions that will minimize a closed box’s cost if it has a square base and top, if its volume is 32 cubic meters, and if the cost per square meter for the top, bottom, and sides respectively are $.03, $.04, and $.05.

(solve step by step please!)

To determine the outside dimensions that will minimize a closed box's cost, follow these steps:

Step 1: Understand the problem:
The problem states that we need to find the dimensions of a closed box with a square base and top that will minimize the cost. The volume of the box is given as 32 cubic meters, and the cost per square meter for the top, bottom, and sides are provided.

Step 2: Define the variables:
Let's define the variables based on the given information:
- Let x be the length of each side of the square base (in meters).
- Let h be the height of the box (in meters).
- Let C be the total cost of the box.

Step 3: Determine equations:
We need to establish relationships between the variables, the volume, and the cost. Since the box has a square base and top, the length, and width of the top and bottom will be x. The area of each side will be x * h.

The relationship between the volume and dimensions is given by:
Volume = length * width * height
32 = x * x * h
32 = x^2 * h ------ equation (1)

The total cost of the box can be calculated by adding the cost of the top, bottom, and all the sides:
C = cost of top + cost of bottom + cost of sides
C = 2 * (x * x) * 0.03 + 2 * (x * h) * 0.05
C = 0.06 * x^2 + 0.1 * x * h ------ equation (2)

Step 4: Express the cost equation in terms of a single variable:
As we want to find the dimensions that minimize the cost, we need to express the cost equation in terms of a single variable. In this case, we can express it in terms of h.

Substitute equation (1) into equation (2):
C = 0.06 * x^2 + 0.1 * x * h
C = 0.06 * x^2 + 0.1 * x * (32 / x^2) ------ substitute h with (32 / x^2)

Simplifying:
C = 0.06 * x^2 + 3.2 / x ------ equation (3)

Step 5: Find the minimum cost:
To find the minimum cost, we need to differentiate equation (3) with respect to x and set it to zero.

dC/dx = 0.12 * x - 3.2 / x^2 ------ differentiate equation (3) with respect to x

Setting dC/dx = 0:
0.12 * x - 3.2 / x^2 = 0

Simplifying and multiplying through by x^2:
0.12 * x^3 - 3.2 = 0

0.12 * x^3 = 3.2

x^3 = 3.2 / 0.12

x^3 = 26.67

Taking the cube root of both sides:
x = ∛26.67

x ≈ 3.029 ------ equation (4)

Step 6: Find the height:
Substitute the value of x from equation (4) into equation (1) to find the height (h):

32 = (3.029)^2 * h

h = 32 / (3.029)^2

h ≈ 3.567 ------ equation (5)

Step 7: Find the outside dimensions:
The outside dimensions are determined by the length of each side of the square base (x) and the height (h). From equation (4) and equation (5), the dimensions of the closed box that minimize the cost are approximately:
- Length of each side of the square base (x) ≈ 3.029 meters
- Height (h) ≈ 3.567 meters

Therefore, the outside dimensions that will minimize the closed box's cost are approximately 3.029 meters by 3.029 meters by 3.567 meters.