A rectangular box with a square base of length x and height h is to have a volume of 20ft^3. The cost of the top and bottom of the box is 20 cents per square foot and the cost of the sides is 8 cents per square foot. Express the cost of the box in terms of:
The variable x and h
The variable x only
The variable h only
Approximate the dimensions of the box that will minimize the cost
clearly
x^2 h = 20 ---> h = 20/x^2 OR x = √(20/h)
so you have :
cost = base + 4 sides
= 20x^2 + 8(4xh)
to get the other two,
replace h with 20/x^2 to get it in terms of x
or
replace the x with √(20/h) to get it in terms of h
To find the cost of the box in terms of the variables x and h, we need to calculate the surface area of each component of the box and multiply it by the corresponding cost per square foot.
1. The cost of the top and bottom of the box:
The area of each square top/bottom is x^2, so the total area for both is 2 * (x^2). Therefore, the cost for the top and bottom would be (2 * (x^2)) * 0.20.
2. The cost of the sides of the box:
The area of each side is x * h, and since there are four sides, the total area of the sides is 4 * (x * h). Therefore, the cost for the sides would be (4 * (x * h)) * 0.08.
Adding the cost of the top/bottom and the cost of the sides, we can express the total cost of the box in terms of x and h as follows:
Cost(x, h) = (2 * (x^2) * 0.20) + (4 * (x * h) * 0.08)
Now, if we want to express the cost of the box in terms of only the variable x, we can consider the value of h as a constant. By substituting h with a specific value or considering a range of possible values, we can simplify the equation:
Cost(x) = (2 * (x^2) * 0.20) + (4 * (x * h_constant) * 0.08)
Similarly, if we want to express the cost of the box in terms of only the variable h, we can consider the value of x as a constant:
Cost(h) = (2 * (x_constant^2) * 0.20) + (4 * (x_constant * h) * 0.08)
To approximate the dimensions of the box that will minimize the cost, we can use calculus. We need to find the critical points by taking the derivative and solving for x and h. Then, we can compare the costs at these critical points to identify the minimum cost. However, since you haven't provided any specific values for h or any constraints, we cannot determine the exact dimensions that will minimize the cost.