To minimize the cost of wood, we need to find the dimensions of the jewelry box that result in the least amount of wood being used.
Let's assume that the dimensions of the square base are x by x (since it's a square) and the height is h.
To find the volume of the jewelry box, we'll use the formula for the volume of a rectangular prism: V = length \times width \times height. In this case, the length (x), width (x), and height (h) are all equal.
Given that the volume of the jewelry box is 4000 cm^3, we have the equation:
4000 = x \times x \times h
Simplifying, we get:
4000 = x^2 \times h
Next, to find the cost of the wood, we need to determine the surface area of each side of the jewelry box.
The top and bottom have the same dimensions, so each has an area of x \times x = x^2.
There are four sides to the jewelry box, so we have a total of four side surfaces, each with an area of x \times h.
The total surface area of the jewelry box is:
2(x^2) + 4(xh) = 2x^2 + 4xh
Given that the wood for the top and bottom costs $20/m^2, and the wood for the sides costs $30/m^2, we can calculate the cost of the wood as follows:
Cost = (2x^2 + 4xh) \times ($20/m^2) + (2x^2) \times ($30/m^2)
To minimize the cost, we need to differentiate the cost equation with respect to one of the variables (x or h) and set the derivative equal to zero, then solve for the variable.
Differentiating the cost equation with respect to x gives:
dCost/dx = 4x + 4h = 0
Differentiating the cost equation with respect to h gives:
dCost/dh = 4x = 0
Setting both derivatives equal to zero, we have:
4x + 4h = 0 (Equation 1)
4x = 0 (Equation 2)
From Equation 2, we know that x = 0. However, since we're dealing with the dimensions of a physical object, x cannot be zero. Therefore, we can ignore Equation 2.
Solving Equation 1 for h, we get:
4h = -4x
h = -x
Now, substituting this value of h into the volume equation:
4000 = x^2 \times h
4000 = x^2 \times (-x)
4000 = -x^3
Simplifying, we find:
x^3 = -4000
Taking the cube root of both sides, we get:
x = -16.85
Since we're working with physical dimensions, the length cannot be negative. Therefore, we disregard the negative value.
Thus, the only possible solution is x ≈ 16.85.
Now, substituting this value of x back into the equation for h:
h = -x
h ≈ -16.85
Again, since the dimensions must be positive, we disregard the negative value.
Therefore, the dimensions that minimize the cost of the wood for a volume of 4000 cm^3 are approximately x = 16.85 cm and h = 16.85 cm.