A box with a square base and no top is to be built with a volume of 4000 in3. Find the dimensions of the box (Length, Width, Height) that requires the least amount of material. How much material is required at the minimum?
see related questions below
To solve this problem, we need to find the dimensions (length, width, and height) of the box that will minimize the amount of material required. Let's break it down step by step:
Step 1: Identify the variables
Let's assign variables to the dimensions of the box: length (L), width (W), and height (H). We need to find the values of L, W, and H that minimize the material usage.
Step 2: Write the formula for the volume of the box
The volume of the box is given as 4000 in3. Since the box has a square base and no top, the formula for the volume is V = L * W * H.
Step 3: Substitute the known values
Substituting the given volume of 4000 in3, we have:
4000 = L * W * H
Step 4: Express one variable in terms of the others
To solve for one variable in terms of the others and minimize material usage, we will express one variable in terms of the other two. In this case, let's express the height (H) in terms of the length (L) and width (W).
Divide both sides of the equation by L * W:
4000 / (L * W) = H
Step 5: Substitute H back into the volume formula
Now that we have expressed H in terms of L and W, we can substitute it back into the volume formula:
V = L * W * H
V = L * W * (4000 / (L * W))
V = 4000
Step 6: Simplify the formula
Simplifying the formula further, we get:
V = 4000
Since the volume of the box will always be equal to 4000 in3, the dimensions of the box do not affect the volume or the amount of material required. Therefore, any dimensions that satisfy the volume requirement (e.g., a box with dimensions 10 in x 10 in x 40 in) will require the same amount of material.
In conclusion, the dimensions of the box (length, width, and height) do not affect the amount of material required.