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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1 answer

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:

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