Answer :
We don't necessarily need a graphing calculator to solve this problem. Here's how we can find the volume that maximizes the box:
- Define the Variables:
Let x represent the side length of the squares cut from the corners (in inches).
- Dimensions of the Box:
After cutting the squares, the remaining length of the metal sheet will be 3 - 2x (length of the rectangle minus twice the square side).
Similarly, the remaining width will be 8 - 2x (width of the rectangle minus twice the square side).
- Volume of the Box:
The box will be open at the top, so we only need to consider the length, width, and height.
Since the metal is folded upwards, the height of the box will be the same as the side length of the cut squares (x).
Therefore, the volume (V) of the box is: V = (length) x (width) x (height) V = (3 - 2x) x (8 - 2x) x (x)
- Maximize the Volume:
To maximize the volume, we can rewrite the expression for V and complete the square. However, there's a shortcut...
Shortcut for Maximization:
Notice that the volume formula (V) has a negative coefficient for the x term squared (-2x^2). A negative coefficient in front of the squared term means the function will have a maximum point.
Since the volume cannot be negative (it's the product of dimensions), the maximum volume will occur when the squared term (x^2) is minimized, which happens when x = 0.
- Verify with Limits (Optional):
While the shortcut suggests x = 0 maximizes the volume, it's good practice to verify this with limits. As x approaches positive or negative infinity, the volume term with x^2 dominates, and the volume approaches negative infinity.
However, this verification isn't strictly necessary as the physical limitations (x cannot be negative) already imply the maximum volume occurs at x = 0.
- Maximum Volume:
If we substitute x = 0 into the volume formula (V), we get: V = (3 - 2(0)) x (8 - 2(0)) x (0) V = 3 x 8 x 0 (since the box has no height with x = 0, the volume becomes 0)
This might seem counterintuitive, but it makes sense. Cutting squares with a side length of 0 from the corners wouldn't remove any material, resulting in a box with no height and zero volume.
- Practical Maximum Volume:
In reality, the manufacturer can't cut squares with zero side length. There will always be a minimum size for the square that can be cut.
To find the practical maximum volume, we need to consider the limitations of the cutting process.
For example, if the minimum square size that can be cut is 1 inch (x = 1), the remaining dimensions become:
Length = 3 - 2(1) = 1 inch
Width = 8 - 2(1) = 6 inches
Height = 1 inch (square side) The volume (V) in this case would be: V = 1 x 6 x 1 = 6 cubic inches
Therefore, the practical maximum volume achievable depends on the minimum possible side length for the squares cut from the corners.
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