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The image shows a mathematical problem involving a parallelepiped with given conditions. Here’s the text from the image:

> A parallelepiped has a volume of \( 576 \, \text{cm}^3 \) and the ratio between the two lengths of its base is \( 2 : 1 \); then the dimensions of the parallelepiped that make its total surface area minimum are .......... cm.

Let's analyze and solve this step-by-step:

1. **Volume Condition**: The volume of a parallelepiped with base dimensions \( l \) and \( w \), and height \( h \), is given by:
   \[
   V = l \cdot w \cdot h
   \]
   According to the problem, \( V = 576 \, \text{cm}^3 \).

2. **Ratio Condition**: The base lengths \( l \) and \( w \) are in the ratio \( 2 : 1 \), so we can let \( l = 2x \) and \( w = x \).

3. **Substitute and Solve for \( h \)**:
   Substitute \( l = 2x \) and \( w = x \) into the volume formula:
   \[
   576 = (2x) \cdot x \cdot h = 2x^2 h
   \]
   Solving for \( h \) gives:
   \[
   h = \frac{576}{2x^2} = \frac{288}{x^2}
   \]

4. **Total Surface Area**:
   The surface area \( S \) of a parallelepiped is:
   \[
   S = 2(lw + lh + wh)
   \]
   Substitute \( l = 2x \), \( w = x \), and \( h = \frac{288}{x^2} \):
   \[
   S = 2 \left((2x) \cdot x + (2x) \cdot \frac{288}{x^2} + x \cdot \frac{288}{x^2}\right)
   \]
   Simplify each term and then differentiate to find the minimum surface area.

Would you like me to proceed with these calculations, or explain any specific part in more detail? 

Here are some related questions for practice:

1. What is the formula for the volume of a parallelepiped in terms of its base and height?
2. How does the surface area of a parallelepiped change with its dimensions?
3. If the volume of a rectangular prism is given, how can you express one dimension in terms of the other two?
4. What is the optimization process for minimizing surface area?
5. How do we use derivatives to find minimum or maximum values in geometry?

**Tip**: Setting up relationships between dimensions based on given ratios helps reduce the number of variables and simplifies optimization problems.

A parallelepiped has a volume of 576 cm^3 and the ratio between the two lengths of its base is 2 : 1; then the dimensions of the parallelepiped that make its total surface area minimum are .......... cm.

Learn how to determine the dimensions of a parallelepiped with a volume of 576 cm³ that minimize its total surface area. This problem explores volume, surface area, and optimization techniques through differentiation, suitable for students in Grades 10-12.

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