A jewelry box with a square base is to be built with copper plated sides, nickel plated bottom and top and a volume of 40cm^3. if nickel plating costs $2 per cm^2 and copper plating costs $1 per cm^2, find the dimensions of the box to maximize the cost of the materials?
To find the dimensions of the box that maximize the cost of the materials, we need to determine the dimensions of the square base and the height of the jewelry box. Let's assume the side length of the square base is "x" and the height is "h".
The volume of the jewelry box is given as 40 cm^3. Since the base is square, the area of the square base is x^2 cm^2, and the volume can be calculated by multiplying the area of the base by the height:
Volume = Area of base * Height
40 cm^3 = x^2 * h
Now, let's calculate the surface area of the box. The four vertical sides (copper plated) have a combined area of 4 * x * h cm^2. The top and bottom (nickel plated) have an area of 2 * x^2 cm^2.
The total cost of the materials can be calculated by multiplying the surface area by the cost per cm^2:
Cost of Copper Plating = Surface area of copper plated sides * Cost per cm^2
= 4 * x * h * $1/cm^2
Cost of Nickel Plating = Surface area of top and bottom * Cost per cm^2
= 2 * x^2 * $2/cm^2
Total Cost = Cost of Copper Plating + Cost of Nickel Plating
= 4 * x * h + 2 * x^2
To maximize the cost of the materials, we need to find the maximum value of the Total Cost function. Let's differentiate it with respect to "x" and set it equal to zero:
d(Total Cost)/dx = d(4*x*h + 2*x^2)/dx
= 4*h + 4*x
Setting it equal to zero:
4*h + 4*x = 0
4*h = -4*x
h = -x
Since the height cannot be negative, it means x should be equal to zero. However, a jewelry box with zero dimensions is not possible, so there must be an error.
Thus, we will use the volume equation to solve for the height in terms of x:
40 cm^3 = x^2 * h
h = 40/x^2
Substituting this value back into the Cost equation:
Total Cost = 4 * x * (40/x^2) + 2 * x^2
= 160/x + 2 * x^2
To find the value of x that maximizes the Total Cost, we can differentiate the Cost function with respect to x and set it equal to zero:
d(Total Cost)/dx = d(160/x + 2*x^2)/dx
= -160/x^2 + 4*x
Setting it equal to zero:
-160/x^2 + 4*x = 0
-160 + 4*x^3 = 0
4*x^3 = 160
x^3 = 40
x ≈ 3.48 cm
Plugging this value of x back into the equation for h:
h = 40/x^2
h ≈ 1.15 cm
Therefore, the dimensions that maximize the cost of the materials are approximately:
Length of side of square base (x) ≈ 3.48 cm
Height (h) ≈ 1.15 cm
To maximize the cost of materials, we need to maximize the surface area of the box while keeping its volume constant.
Let's start by considering the dimensions of the box. Let's assume the length of one side of the square base is "x". Since the box has a square base, its width will also be "x". The height of the box will be denoted as "h".
The volume of a rectangular box is given by the formula: Volume = length * width * height. In this case, the volume is given as 40 cm^3. So we have the equation:
x * x * h = 40
Next, let's calculate the surface area of the box. The surface area of the box is the sum of the areas of all its sides. The nickel plating covers the bottom and top of the box, so the area covered by nickel is 2 times the area of the square base. The copper plating covers the four sides of the box, so the area covered by copper is 4 times the area of the square base.
The area of the square base is given by: area = side * side. Therefore, the area of each side of the square base is x * x.
The total surface area of the box is then:
Surface Area = 2 * (nickel area) + 4 * (copper area)
= 2 * (x * x) + 4 * (x * x)
= 2x^2 + 4x^2
= 6x^2
Now, we can express the cost of materials in terms of the surface area. The cost of nickel plating is $2 per cm^2, and the cost of copper plating is $1 per cm^2.
Cost = (nickel cost) + (copper cost)
= 2 * (nickel area) + 1 * (copper area)
= 2 * (2x^2) + 1 * (4x^2)
= 4x^2 + 4x^2
= 8x^2
Now, we want to maximize the cost, which means we need to find the maximum value of the cost function in terms of "x". To do this, let's differentiate the cost function with respect to "x" and set it equal to zero:
d(Cost)/dx = 16x
16x = 0
x = 0
Since the length of a side cannot be zero, we can disregard this result. Therefore, there is no maximum cost for this problem.
However, if we want to find the dimensions of the box that maximize the cost given the constraints, we can substitute the value of "x" back into the equation for the volume:
x * x * h = 40
We can solve this equation for "h" to find the height of the box.
not sure why you'd want to maximize the costs, but the initial steps are the same.
Box has sides of x, and height y. We know that
x^2y = 40, so y = 40/x^2
So, the cost is
c(x) = 2*2*x^2 + 1*4*xy = 4x^2 + 160/x
I think you can see that there is no maximum value, so now just find the minimum, where dc/dx = 0