Height: So the corners are cut out from a square sheet, which means the height will be the same as the length of each cut-out square. Let's call the length of each cut-out square x. When the flaps are bent up, the height is x as well.
Length: After the corners are cut out, the remaining sheet will have a length of 7 - 2x (two corners are cut out from each end). When folded, this length will be the same as the length of the box.
Width: Similarly, the width of the box will be the same as the width of the remaining sheet, which is also 7 - 2x.
To maximize the volume, we need to maximize the length, width, and height of the box. So let's differentiate their product with respect to x and find the value of x that yields the maximum volume.
V = (7 - 2x)(7 - 2x)x
Differentiating with respect to x:
dV/dx = 2(7 - 2x)(-2x) + (7 - 2x)(7 - 2x)
Setting dV/dx = 0:
2(7 - 2x)(-2x) + (7 - 2x)(7 - 2x) = 0
Expanding and simplifying:
-4x(7 - 2x) + (7 - 2x)(7 - 2x) = 0
-28x + 8x^2 + 49 - 28x + 4x^2 = 0
12x^2 - 56x + 49 = 0
Solving this quadratic equation, we get two values for x:
x = 1.71 and x = 2.88
Since the dimensions cannot be negative, we discard x = 2.88.
Therefore, the largest box that can be made has the following dimensions:
Height: 1.71 inches
Length: (7 - 2(1.71)) = 3.58 inches
Width: (7 - 2(1.71)) = 3.58 inches
Now that we have the dimensions, we can enjoy a perfectly sized box for all our tin-related storage needs. Just try not to cut yourself on those flaps!