To determine the values of \( x \) that make the volume \( V = 0 \), we need to first establish the volume formula for the box that is created by cutting squares of side length \( x \) from each corner of a 16-inch by 16-inch piece of tin.
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Dimensions of the Box After cutting squares of length \( x \) from each corner, the dimensions of the box will be:
- Length: \( 16 - 2x \)
- Width: \( 16 - 2x \)
- Height (After folding up the sides): \( x \)
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Volume of the Box The volume \( V \) of the box can be expressed as: \[ V(x) = \text{Length} \times \text{Width} \times \text{Height} = (16 - 2x)(16 - 2x)(x) = (16 - 2x)^2 \cdot x \]
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Setting the Volume to Zero To find the values of \( x \) such that \( V(x) = 0 \), we can set the volume equation to zero: \[ (16 - 2x)^2 \cdot x = 0 \]
This equation can be satisfied in two ways:
- \( (16 - 2x)^2 = 0 \)
- \( x = 0 \)
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Solving for \( x \)
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For \( (16 - 2x)^2 = 0 \): \[ 16 - 2x = 0 \] \[ 2x = 16 \] \[ x = 8 \]
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The other case is straightforward: \[ x = 0 \]
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Results The solutions for \( x \) that make the volume \( V \) equal to zero are: \[ x = 0 \quad \text{and} \quad x = 8 \]
Therefore, \( x \) can be:
- \( x = 0 \)
- \( x = 8 \)
Thus, the values of \( x \) that make \( V = 0 \) are:
\[ \text{Answer: } 0, 8 \]