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To solve the problem of calculating the surface area of the smaller container, we need to understand that the two containers are mathematically similar. This means that their dimensions are proportional, and thus the ratios of corresponding linear dimensions, surface areas, and volumes follow a specific relationship.

We are given the following information:
- Larger container capacity = 64.8 liters
- Larger container surface area = 0.792 m²
- Smaller container capacity = 37.5 liters

### Steps to solve:

1. **Volume Ratio**: 
   The volumes (capacities) of two similar shapes are proportional to the cube of their corresponding linear dimensions. Thus, the ratio of their volumes will be:
   \[
   \frac{\text{Volume of smaller container}}{\text{Volume of larger container}} = \left(\frac{\text{Linear dimension of smaller container}}{\text{Linear dimension of larger container}}\right)^3
   \]
   Therefore,
   \[
   \left(\frac{\text{Linear dimension of smaller container}}{\text{Linear dimension of larger container}}\right) = \sqrt[3]{\frac{\text{Volume of smaller container}}{\text{Volume of larger container}}}
   \]

2. **Surface Area Ratio**:
   The surface areas of two similar shapes are proportional to the square of their corresponding linear dimensions. Hence, the ratio of their surface areas is:
   \[
   \frac{\text{Surface area of smaller container}}{\text{Surface area of larger container}} = \left(\frac{\text{Linear dimension of smaller container}}{\text{Linear dimension of larger container}}\right)^2
   \]

3. **Calculations**:
   First, calculate the ratio of the linear dimensions:
   \[
   \frac{\text{Linear dimension of smaller container}}{\text{Linear dimension of larger container}} = \sqrt[3]{\frac{37.5}{64.8}}
   \]

   Then, use this ratio to find the ratio of the surface areas:
   \[
   \frac{\text{Surface area of smaller container}}{\text{Surface area of larger container}} = \left(\sqrt[3]{\frac{37.5}{64.8}}\right)^2
   \]

   Finally, multiply the surface area of the larger container by this ratio to get the surface area of the smaller container.

Let me calculate this for you.The surface area of the smaller container is approximately **0.55 m²**.

Would you like further clarification or details on any part of this solution?

Here are 5 related questions you might find useful:
1. How would the surface area change if the containers were not mathematically similar?
2. What is the relationship between volume and surface area for different shapes?
3. How can we calculate the surface area of a container with more complex shapes?
4. How would the answer change if we had different units for volume or surface area?
5. How do scale factors affect other physical properties, like weight?

**Tip**: In similar figures, ratios of areas are the square of the ratio of linear dimensions, while ratios of volumes are the cube of the ratio.

Two containers are mathematically similar. The larger container has a capacity of 64.8 liters and a surface area of 0.792 m². The smaller container has a capacity of 37.5 liters. Calculate the surface area of the smaller container.

Learn how to calculate the surface area of a smaller container given that it is mathematically similar to a larger container. Understand the relationship between the volumes and surface areas using similarity principles. Suitable for students in Grades 9-11.

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