A box is constructed out of two different types of metal. The metal for the top and bottom, which are both square, costs $4 per square foot and the metal for the sides costs $4 per square foot. Find the dimensions that minimize cost if the box has a volume of 50 cubic feet.
Length of base x=........
Height of side z= .....
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To minimize the cost of the box, we need to find the dimensions that minimize the total surface area of the box.
Let's assign the following variables:
Length of the base (top and bottom): x
Height of the side: z
The volume of the box is given as 50 cubic feet, so we can write the equation:
Volume of the box = Length x Width x Height
50 = x * x * z
Now, to find the total surface area, we need to consider the top, bottom, and four sides of the box.
Area of the top and bottom = x * x
Area of each side = x * z
Total surface area = 2(Area of top and bottom) + 4(Area of each side)
Total surface area = 2(x * x) + 4(x * z)
Next, we need to express the cost in terms of the surface area. Since the cost of the metal is $4 per square foot, the cost of the top and bottom is 4(x * x), and the cost of each side is 4(x * z).
Total cost = Cost of top and bottom + Cost of each side
Total cost = 4(x * x) + 4(x * z)
Now, we need to express z in terms of x using the volume equation:
z = 50/(x * x)
Substituting this value of z into the equation for the total cost:
Total cost = 4(x * x) + 4(x * (50/(x * x)))
Simplifying the equation:
Total cost = 4(x * x) + 4(50/x)
To minimize the cost, we can take the derivative of the cost function with respect to x:
d(Total cost)/dx = 8x - 200/x^2
Setting the derivative equal to zero to find the critical points:
8x - 200/x^2 = 0
8x = 200/x^2
8x^3 = 200
x^3 = 25
x = ∛(25)
x ≈ 2.924
Since the dimensions cannot be negative, we ignore the negative values.
Therefore, the length of the base (x) that minimizes the cost is approximately 2.924 feet.
To find the height of the side (z), we substitute this value of x back into the volume equation:
z = 50/(x * x)
z = 50/(2.924 * 2.924)
z ≈ 6.200
Therefore, the height of the side (z) that minimizes the cost is approximately 6.200 feet.