A box can be formed by cutting squares out of each corner of a piece of tin and folding the​ "tabs" up. Suppose the piece of tin is 16 inches by 16 inches and each side of the square that is cut out has length x.

Complete parts ​(a) through ​(e).
(e) For which of these values of x does a box exist if squares of length x are cut out and the tabs folded​ up?
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.

1 answer

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.

  1. The piece of tin has dimensions \( 16 \) inches \( \times \) \( 16 \) inches.
  2. 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.
  3. 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.