Question

A cardboard box is formed by cutting squares from the corners of a 12 in. by 8 in. rectangular piece of cardboard, folding up the sides (along the dotted lines), and taping the corners as shown.



Part A
Formulate a model function for the problem situation, where V(w) is the volume in cubic inches of the box and w is the width in inches of the squares cut out of the rectangular piece of cardboard.

Part B
Create a graph of the function over the appropriate mathematical domain and range, and create a graph of the model function over an appropriate domain and range for the problem situation.




Part C
What are the roots of the graph of V(x) over the domain of all real numbers? What is significant about the roots in terms of this problem situation?

Part D
What is the maximum volume possible for this box and what would be the side width of the cut out square? Round the volume to the nearest tenth of a cubic inch and the width of the cut out square to the nearest hundredth of an inch.

Answer 1

The maximum volume of the box is the highest volume the box can have.

• The model of the function is: $\mathbf{V(w) = (4 + w) \times w \times \frac{8 - w}{2}}$
• The roots of V(x) are: $\mathbf{x = 0,4,6}$.
• The maximum volume is: 67.60 cubic inches, and the width of the cut-out is 1.569 inch

(a) A model function V(w)

The dimension of the cardboard is given as:

$\mathbf{l =12}$

$\mathbf{w =8}$

Assume the cut-out is x.

So, we have:

$\mathbf{l =12 - 2x}$

$\mathbf{w =8 - 2x}$

$\mathbf{h = x}$

Make x the subject in $\mathbf{w =8 - 2x}$

$\mathbf{x = \frac{8 - w}{2}}$

The volume of the box is calculated as:

$\mathbf{V = lwh}$

Substitute expressions for l and h

$\mathbf{V = (12 - 2x) \times w \times x}$

Substitute $\mathbf{x = \frac{8 - w}{2}}$

$\mathbf{V = (12 - 2\times (\frac{8 - w}{2})) \times w \times \frac{8 - w}{2}}$

$\mathbf{V = (12 - (8 - w)) \times w \times \frac{8 - w}{2}}$

$\mathbf{V = (12 - 8 + w) \times w \times \frac{8 - w}{2}}$

$\mathbf{V = (4 + w) \times w \times \frac{8 - w}{2}}$

So, we have:

$\mathbf{V(w) = (4 + w) \times w \times \frac{8 - w}{2}}$

Hence, the model of the function is: $\mathbf{V(w) = (4 + w) \times w \times \frac{8 - w}{2}}$

(b) The graph of the function and the model function

In (a), we have:

$\mathbf{V = (12 - 2x) \times w \times x}$

Substitute $\mathbf{w =8 - 2x}$

$\mathbf{V = (12 - 2x) \times (8 -2x) \times x}$

So, we have:

$\mathbf{V(x) = (12 - 2x) \times (8 -2x) \times x}$

See attachment for the graphs of V(x) and V(w)

(c) The roots of V(x)

From the graph of V(x), the roots are:

$\mathbf{x = 0,4,6}$

These values represent the possible cut-outs from the cardboard

(d) The maximum volume

From the graphs of V(x) and V(w), the maximum volume is:

$\mathbf{Maximum = 67.60}$

And the width of the cut-out is:

$\mathbf{Width= 1.569}$

Hence, the maximum volume is: 67.60 cubic inches, and the width of the cut-out is 1.569 inch

Read more about volumes at:

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