To determine for which values of \( x \) a box can be formed by cutting squares with length \( x \) from each corner of a 16-inch by 16-inch piece of tin, we need to analyze the constraints created by the dimensions of the tin.
- The piece of tin has dimensions \( 16 \) inches \( \times \) \( 16 \) inches.
- When squares of length \( x \) are cut out from each corner, the dimensions of the base of the resulting box will be reduced by \( 2x \) in both the length and width.
- Therefore, the dimensions of the base of the box after folding up the sides will be \( (16 - 2x) \times (16 - 2x) \).
For a valid box to exist, the dimensions of the base must be positive. Hence, we need:
\[ 16 - 2x > 0 \]
Solving this inequality:
\[ 16 > 2x \] \[ 8 > x \]
Thus, the maximum value for \( x \) is \( x < 8 \).
Additionally, \( x \) must be non-negative since we cannot cut out a negative square:
\[ x \geq 0 \]
Combining these inequalities, we have:
\[ 0 \leq x < 8 \]
This means \( x \) can take on any value within the range from \( 0 \) to \( 8 \) (not including 8).
Now, the options provided in part (e) appear to represent conclusions on which values \( x \) permits the formation of a box:
A. For all the values of \( x \).
B. For neither of the values of \( x \).
C. For the larger value of \( x \).
D. For the smaller value of \( x \).
Given our analysis, a box can exist for any value \( x \) such that \( 0 \leq x < 8 \). Therefore, the correct answer is:
A. For all the values of x.
Assuming the term "values" refers to all valid values in the range \( 0 \leq x < 8 \). If it's more specific and the question intends only specific values, we would need clarification on what "larger" and "smaller" values refer to. However, considering \( x \) must be within the given constraints, A is the most inclusive answer that applies to all possible \( x \) values.