An open box (has a bottom but no top) needs to be constructed which has a volume of 500 cu. in. The bottom of the box is to be a square. Let x be one of the sides of the bottom. Express the area of the four sides and bottom of box as a function of x alone. What value of x will make the total area of the 4 sides and the bottom of the box as small as possible? (that is, gives the box that uses the least amount of material to build and has a volume of 500).
Please help and explain. I have an exam shortly and a similar question will be on it.
If the base is x by x, let the height be y
Volume = (x^2)y
x^2y = 500
y = 500/x^2
surface area = x^2 + 4xy
= x^2 + 4x(500/x^2 = x^2 + 2000/x
d(surface area)/dx = 2x - 2000/x^2
= 0 for a max/min of surface area
2x - 2000/x^2 = 0
x^3 = 1000
x = 10
To find the area of the four sides and the bottom of the box as a function of x alone, we need to calculate the areas of each individual side.
The bottom of the box is a square, so its area is x^2.
The four sides of the box are identical rectangles, and they have the same height as the bottom of the box, which is x. The length of each side can be found by dividing the volume of the box (500 cu. in.) by the area of the base (x^2).
Since the volume of a rectangular prism is given by V = length × width × height, in this case, we have V = x^2 × x = x^3.
Therefore, the length of each side is (500/x^2)^(1/3).
The four sides have the same dimensions, so each side has an area equal to (500/x^2)^(1/3) × x.
Now we can find the total area by summing the area of the bottom (x^2) and the area of the four sides. Writing it as a function of x, we have:
A(x) = x^2 + 4(500/x^2)^(1/3) × x
Next, we need to find the value of x that minimizes this function to determine the value that will result in the box using the least amount of material.
To find the minimum value of A(x), we can take the derivative of A(x) with respect to x and set it equal to zero.
dA/dx = 2x - 4(500/x^2)^(1/3) + 4(500/x^2)^(-2/3) × (1/3) × x^(-3)
Setting dA/dx = 0, we can solve for x.
2x - 4(500/x^2)^(1/3) + 4(500/x^2)^(-2/3) × (1/3) × x^(-3) = 0
Simplifying and rearranging the equation (you may multiply everything through by 3x^2 for simplification), we get:
6x^3 - 12(500/x^2)^(1/3) + 4(500/x^2)^(-2/3) = 0
Unfortunately, this equation is not easily solvable algebraically. However, you can utilize numerical methods such as graphing or using a calculator to find the value of x that minimizes the function A(x).
Once you find the value of x, substitute it back into the equation for A(x) to get the minimum area.