To find the volume of the box formed by cutting squares out of each corner of an 8-inch by 8-inch piece of tin and folding the "tabs" up, we can break down each part as follows:
Part 1
(a) The height of the box is the same as the length of the side of the squares that were cut out. Therefore, if you cut out squares of side length \( x \):
The height of the box is:
\[ x \text{ inches} \]
Part 2
(b) When you cut out the squares and fold up the sides, the dimensions of the base of the box are reduced on each side by twice the length of the cut-out square (because squares are cut from both sides of each dimension).
Given that the original dimensions of the tin are 8 inches by 8 inches:
The dimensions of the base are:
\[ (8 - 2x) \text{ inches by } (8 - 2x) \text{ inches} \]
Part 3
(c) To find the volume \( V \) of the box, use the volume formula for a rectangular prism: \[ V = l \cdot w \cdot h \] where \( l \) is the length, \( w \) is the width, and \( h \) is the height.
From Part 1 and Part 2, we have:
- \( h = x \)
- \( l = 8 - 2x \)
- \( w = 8 - 2x \)
Substituting these values into the volume formula, we get: \[ V = (8 - 2x) \cdot (8 - 2x) \cdot x \]
This simplifies to: \[ V = (8 - 2x)^2 \cdot x \]
Now expand \( (8 - 2x)^2 \): \[ (8 - 2x)^2 = 64 - 32x + 4x^2 \]
Thus, the volume can be expressed as: \[ V = (64 - 32x + 4x^2) \cdot x \]
Finally, multiplying through by \( x \): \[ V = 4x^3 - 32x^2 + 64x \]
The volume of the box is: \[ V = 4x^3 - 32x^2 + 64x \]
This is the equation that represents the volume of the box based on the side length \( x \) of the squares cut out of each corner.
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