The problem is asking to find a formula for the amount of packing material, $M(x)$, needed to surround a cardboard box with given dimensions, allowing for space between the box and a wooden crate.

The dimensions of the box are:

• Length: $x$ feet
• Width: $x + 2$ feet
• Depth: $x - 1$ feet

There must be:

• A 0.5-foot space on the front, back, and sides.
• A 1-foot space on the top and bottom.

Step 1: Dimensions of the crate

The crate must be larger than the box to accommodate the packing material. So, we add the required space around each dimension:

• Length of the crate: Since there is a 0.5 ft space on the front and back, the total added space is 1 foot. Thus, the length of the crate is: $\text{Length of crate} = x + 1$

• Width of the crate: Similar to the length, there is a 0.5 ft space on both sides, so the width of the crate is: $\text{Width of crate} = (x + 2) + 1 = x + 3$

• Depth of the crate: For the depth, we account for the 1-foot space at the top and bottom, so the depth of the crate is: $\text{Depth of crate} = (x - 1) + 2 = x + 1$

Step 2: Volume of the crate and the box

The volume of the crate and the box can now be calculated:

• Volume of the crate: $V_{\text{crate}} = (x + 1)(x + 3)(x + 1)$

• Volume of the box: $V_{\text{box}} = x(x + 2)(x - 1)$

Step 3: Formula for the packing material

The amount of packing material $M(x)$ is the difference between the volume of the crate and the volume of the box: $M(x) = V_{\text{crate}} - V_{\text{box}}$

Substituting the expressions for the volumes:

$M(x) = (x + 1)(x + 3)(x + 1) - x(x + 2)(x - 1)$

This is the formula for the amount of packing material needed in terms of $x$.

Would you like further details or a simplification of this formula?

Related Questions:

1. How would the formula change if the spaces around the box were different sizes?
2. What is the packing material required if the box length is 4 feet?
3. How does the volume of the box affect the overall packing material?
4. Can you derive the expanded form of the packing material formula?
5. What happens to the packing material required as the box size approaches zero?

Tip: Remember that when calculating volumes, ensure the units match across all dimensions.