A company needs to packages 80 cubic feet of chemicals in to a plastic container, find the whole number dimensions of a container that would use the least amount of plastic
4x4x5
What’s the equation anyone?
To find the whole number dimensions of a container that would use the least amount of plastic, we need to determine the dimensions that would result in the smallest surface area.
Let's assume the container is in the shape of a rectangular prism.
Let's represent the dimensions of the container as length (L), width (W), and height (H).
Since we want to minimize the amount of plastic used, we need to minimize the surface area of the container.
The formula for the surface area of a rectangular prism is:
Surface area = 2(LW + LH + WH)
We also know that the volume of the container is 80 cubic feet, so:
Volume = L * W * H = 80
To find the dimensions that use the least amount of plastic, we need to minimize the surface area while keeping the volume constant.
There are several approaches to solving this problem, including trial and error, but one common method is to use calculus.
By using calculus, we can find the minimum surface area by finding the critical points of the surface area function. However, since we are looking for whole number dimensions, we can use a more straightforward approach.
Let's find the factors of 80 and see which combination results in the smallest surface area.
The factors of 80 are: 1, 2, 4, 5, 8, 10, 16, 20, 40, 80
We can now try different combinations of these factors to find the one with the smallest surface area.
For example, let's consider the combination of factors (L, W, H) = (2, 5, 8):
Surface area = 2(2*5 + 2*8 + 5*8) = 2(10 + 16 + 40) = 2(66) = 132 square feet
Let's try another combination, (L, W, H) = (4, 4, 5):
Surface area = 2(4*4 + 4*5 + 4*5) = 2(16 + 20 + 20) = 2(56) = 112 square feet
Continuing this process, we can calculate the surface area for each combination and find the one that results in the smallest value.
By comparing the surface areas of these combinations, we can determine the whole number dimensions that would use the least amount of plastic. In this case, the combination (L, W, H) = (4, 4, 5) has the smallest surface area of 112 square feet, so the whole number dimensions of the container would be 4 feet by 4 feet by 5 feet.