Question

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).

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
Use the equation of the volume of the box to find the values of x that make Upper V equals 0.
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Answer 1

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.

1. **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): \( x \)

2. **Volume of the Box**
The volume \( V \) of the box can be expressed as:
\[
V(x) = \text{Length} \times \text{Width} \times \text{Height} = (16 - 2x)(16 - 2x)(x) = (16 - 2x)^2 \cdot x
\]

3. **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:
- \( (16 - 2x)^2 = 0 \)
- \( x = 0 \)

4. **Solving for \( x \)**
- For \( (16 - 2x)^2 = 0 \):
\[
16 - 2x = 0
\]
\[
2x = 16
\]
\[
x = 8
\]

- The other case is straightforward:
\[
x = 0
\]

5. **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:

\[
\text{Answer: } 0, 8
\]