A company needs to package hazardous chemicals in special plastic rectangular prism containers that hold 80 cubic feet. Find the whole number dimensions of the container that would use the least amount of plastic.
To find the dimensions that would use the least amount of plastic, we need to minimize the surface area of the container.
The surface area of a rectangular prism is given by the formula:
Surface Area = 2lw + 2lh + 2wh
Where:
l = length
w = width
h = height
Given that the volume of the container is 80 cubic feet, we have:
Volume = lwh = 80
We want to minimize the surface area, so we want to minimize the sum of the sides of the container. We can rewrite the surface area formula in terms of one variable using the volume equation:
Surface Area = 2lw + 2(80/l) + 2(w(80/l)) = 2l(2w + 80/l)
To minimize the surface area, we need to minimize this expression.
Let's find the derivative of this expression with respect to l:
d(Surface Area)/dl = 2(2w - 80/l^2)
Setting the derivative equal to zero:
2w - 80/l^2 = 0
2w = 80/l^2
l^2 = 80/(2w)
l = sqrt(80/(2w))
l = sqrt(40/w)
To find the whole number dimensions, we can plug in values for w and calculate the corresponding value for l. Let's try w = 2:
l = sqrt(40/2) = sqrt(20) = 4.47 (approx)
Since the dimensions need to be whole numbers, this dimension does not work.
Let's try w = 3:
l = sqrt(40/3) = sqrt(13.33) = 3.65 (approx)
Again, this dimension does not work.
Let's try w = 4:
l = sqrt(40/4) = sqrt(10) = 3.16
This dimension works and is close to a whole number.
Therefore, the dimensions of the container that would use the least amount of plastic are:
Length = 3 ft
Width = 4 ft
Height = 10 ft