A rectangular box is to have a square base and a volume of 50 ft3. The material for the base costs 32¢/ft2, the material for the sides costs 10¢/ft2, and the material for the top costs 26¢/ft2. Letting x denote the length of one side of the base, find a function in the variable x giving the cost (in dollars) of constructing the box.
we've been through several of these box problems now.
Have you some ideas of how to proceed?
I had to miss my last week of Calculus due to some personal things, so I had not been through any lectures of optimization. It hurt me greatly, now I have a few of these that I don't know how to set up at all.
base is side x, height is h, so
hx^2 = 50
h = 50/x^2
cost of the box is
cost of sides + cost of bottom + cost of top
that should get you started.
To find the function in the variable x giving the cost of constructing the box, we need to determine the dimensions of the box first.
Let's start by considering the base of the box. Since the box has a square base, we know that the length of one side of the base is equal to x. Therefore, the area of the base can be calculated as A = x^2.
Next, we determine the height of the box. The volume of the box is given as 50 ft^3, and the volume of a rectangular box is obtained by multiplying its base area by its height. So, we can write the equation as V = A * h, where V represents the volume, A represents the base area, and h represents the height. Substituting the known values, we have 50 = x^2 * h.
Solving for h, we find h = 50 / x^2.
Now that we have the dimensions of the box, we can calculate the cost of constructing it. The total cost consists of three parts: the cost of the base, the cost of the sides, and the cost of the top.
The cost of the base is given as 32¢/ft^2, and since the area of the base is x^2, the cost of the base can be calculated as C_base = 0.32 * x^2.
The cost of the sides is given as 10¢/ft^2, and there are four sides on the box. The total cost of the sides can be calculated as C_sides = 4 * (0.10 * A), where A represents the area of one side. Since the sides have the same height as the box, the height can be used as their length, so C_sides = 4 * (0.10 * x * h) = 0.4 * x * (50 / x^2) = 20 / x.
The cost of the top is given as 26¢/ft^2, and since the area of the top is x^2, the cost of the top can be calculated as C_top = 0.26 * x^2.
Finally, the total cost of constructing the box is the sum of the costs of the base, sides, and top. So, the function C(x) giving the cost of constructing the box in dollars can be expressed as:
C(x) = C_base + C_sides + C_top
= 0.32 * x^2 + 20 / x + 0.26 * x^2
= 0.58 * x^2 + 20 / x
Therefore, the function C(x) = 0.58 * x^2 + 20 / x represents the cost (in dollars) of constructing the box.