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 8 inches by 8 inches and each side of the square that is cut out has length x.

Question

Complete parts ​(a) through ​(e).
8
8
8
8
x
x
Question content area bottom
Part 1
​(a) Write an expression for the height of the box that is constructed.
The height of the box is
  
x inches.
Part 2
​(b) Write an expression for the dimensions of the base of the box that is constructed.
The dimensions of the base of the constructed box are
  
8 minus 2 x inches by
  
8 minus 2 x inches.
Part 3
​(c) Use the formula Upper V equals lwh to find an equation that represents the volume of the box.
The volume of the box is
  
4 x cubed minus 32 x squared plus 64 x.
Part 4
​(d) Use the equation of the volume of the box to find the values of x that make Upper V equals 0.
xequals
  
enter your response here ​(Use a comma to separate answers as​ needed.)

Answer 1

To find the volume and the values of \( x \) that make \( V = 0 \), let’s start by following the steps outlined above.

Part 1

(a) The height of the box after cutting out squares of side \( x \) inches from each corner and folding up the tabs will be:

• Height: \( x \) inches.

Part 2

(b) The dimensions of the base of the box will change as the squares are cut out. Each side of the square cut at the corner reduces the corresponding dimension by 2 times \( x \) (one from each side).

• Base Dimensions: \( (8 - 2x) \) inches by \( (8 - 2x) \) inches.

Part 3

(c) The volume \( V \) of the box can be expressed as a function of its dimensions: \[ V = \text{Length} \times \text{Width} \times \text{Height} = (8 - 2x)(8 - 2x)(x) \] This simplifies to: \[ V = (8 - 2x)^2 \cdot x \] Now, we can expand: \[ (8 - 2x)(8 - 2x) = 64 - 32x + 4x^2 \] So then, multiplying by \( x \): \[ V = x(64 - 32x + 4x^2) = 4x^3 - 32x^2 + 64x \]

Part 4

(d) To find the values of \( x \) that make \( V = 0 \), we can set the volume equation equal to zero: \[ 4x^3 - 32x^2 + 64x = 0 \] We can factor this out: \[ 4x(x^2 - 8x + 16) = 0 \] This gives us: \[ 4x = 0 \quad \text{or} \quad x^2 - 8x + 16 = 0 \] From \( 4x = 0 \), we have \( x = 0 \).

For the quadratic part: \[ x^2 - 8x + 16 = 0 \implies (x - 4)^2 = 0 \implies x = 4. \] Thus, the values of \( x \) that make \( V = 0 \) are: \[ x = 0, 4. \]

Answer

Therefore, the values of \( x \) are \( \boxed{0, 4} \).