To calculate the cost of covering the outside of the box with $4 tiles, we need to determine the surface area of the box first and then multiply it by the cost per square meter ($4). Here's how you can express it step-by-step:

Step 1: Surface area of the box

The box has a length, width, and height. Based on the problem:

• Length = $L$ meters
• Width = $L - 3$ meters
• Height is unspecified, so we'll call it $h$ meters.

The surface area of a rectangular box is calculated using:

$\text{Surface Area} = 2(\text{Length} \cdot \text{Width} + \text{Length} \cdot \text{Height} + \text{Width} \cdot \text{Height})$

Substituting the given values:

$\text{Surface Area} = 2\big(L(L - 3) + Lh + (L - 3)h\big)$

Simplify the expression:

$\text{Surface Area} = 2\big(L^2 - 3L + Lh + (Lh - 3h)\big)$

$\text{Surface Area} = 2\big(L^2 - 3L + 2Lh - 3h\big)$

Step 2: Cost of covering the box

The cost to cover one square meter is $4. Therefore, the total cost is:

$\text{Cost} = 4 \cdot \text{Surface Area}$

Substitute the surface area into the cost formula:

$\text{Cost} = 4 \cdot 2\big(L^2 - 3L + 2Lh - 3h\big)$

Simplify:

$\text{Cost} = 8\big(L^2 - 3L + 2Lh - 3h\big)$

This is the final expression for the total cost of covering the outside of the box in $4 tiles.

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Final Answer:

The cost is:

$\boxed{8\big(L^2 - 3L + 2Lh - 3h\big)}$

Let me know if you'd like me to clarify any part or expand on this!

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Follow-up Questions:

1. How does the surface area of the box change if $h$ is doubled?
2. What is the cost if $L = 5$ meters and $h = 2$ meters?
3. How does the formula simplify if $L = h$?
4. How does the cost change if tiles are $5 each instead of $4?
5. Can you derive the volume of the box using the same $L$, $L - 3$, and $h$?

Tip:

When solving problems with surface areas and costs, always carefully verify the dimensions and units to avoid errors.