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The problem in the image is about finding the dimensions of a rectangular prism (a box) with a volume of \(2 \, \text{m}^3\) that uses the least amount of material. The box has a cover, which implies that we need to consider the surface area including the top.

Let \( l \) be the length, \( w \) be the width, and \( h \) be the height of the box. The volume constraint is:
\[ l \cdot w \cdot h = 2 \]

The surface area \( S \) of the box with a cover is given by:
\[ S = 2(lw + lh + wh) \]

We need to minimize \( S \) subject to the volume constraint \( l \cdot w \cdot h = 2 \).

To solve this optimization problem, we use the method of Lagrange multipliers or directly substitute the volume constraint into the surface area equation and find the critical points.

1. **Express \( h \) in terms of \( l \) and \( w \):**
\[ h = \frac{2}{lw} \]

2. **Substitute \( h \) into the surface area equation:**
\[ S = 2 \left(lw + l\frac{2}{lw} + w\frac{2}{lw}\right) \]
\[ S = 2 \left(lw + \frac{2}{w} + \frac{2}{l}\right) \]

3. **To find the minimum surface area, we need to find the partial derivatives of \( S \) with respect to \( l \) and \( w \), and set them to zero:**

\[ \frac{\partial S}{\partial l} = 2 \left(w - \frac{2}{l^2}\right) = 0 \]
\[ w - \frac{2}{l^2} = 0 \]
\[ w = \frac{2}{l^2} \]

\[ \frac{\partial S}{\partial w} = 2 \left(l - \frac{2}{w^2}\right) = 0 \]
\[ l - \frac{2}{w^2} = 0 \]
\[ l = \frac{2}{w^2} \]

4. **Since \( l = \frac{2}{w^2} \) and \( w = \frac{2}{l^2} \), substituting \( w \) from one equation into the other:**

\[ l = \frac{2}{\left(\frac{2}{l^2}\right)^2} \]
\[ l = \frac{2}{\frac{4}{l^4}} \]
\[ l = \frac{2l^4}{4} \]
\[ l = \frac{l^4}{2} \]
\[ 2 = l^3 \]
\[ l = \sqrt[3]{2} \]

5. **Since \( l = w = h \), we have:**

\[ l = w = h = \sqrt[3]{2} \]

Therefore, the optimal dimensions of the box, which minimize the material used, are:
\[ l = w = h = \sqrt[3]{2} \]

Do you need more details or have any questions? Here are some questions you might consider:

1. How do we apply Lagrange multipliers to this problem?
2. What is the significance of minimizing surface area in practical applications?
3. Can you show more steps in solving the partial derivatives?
4. How do the dimensions change if the volume is different?
5. Can this method be applied to other shapes?

**Tip:** Always check the second derivative to confirm that the critical points correspond to a minimum rather than a maximum or saddle point.

Explore how to optimize the dimensions of a rectangular prism with a cover to minimize material usage, given a volume constraint. Includes detailed steps on using Lagrange multipliers and calculating surface area. Suitable for advanced high school students.

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