The question asks us to calculate the surface area of a smaller container, given that two containers are mathematically similar. Here's the relevant information from the problem:

• Larger container's capacity: $64.8 \, \text{litres}$
• Larger container's surface area: $0.792 \, \text{m}^2$
• Smaller container's capacity: $37.5 \, \text{litres}$

Steps to solve:

1. Relationship of Similar Shapes: When two objects are mathematically similar, their volumes are related by the cube of the scale factor, and their surface areas are related by the square of the scale factor.

   Let the scale factor between the containers be $k$.

   Volume relationship: $\left( \frac{\text{Volume of smaller container}}{\text{Volume of larger container}} \right) = k^3$ $\frac{37.5}{64.8} = k^3$ Calculate $k$: $k^3 = \frac{37.5}{64.8}$ $k = \sqrt[3]{\frac{37.5}{64.8}}$

2. Surface Area Relationship: Since the surface area is related by $k^2$, the surface area of the smaller container will be: $\text{Surface Area of smaller container} = (\text{Surface Area of larger container}) \times k^2$ Now we can compute the values.

Let me calculate these values.The surface area of the smaller container is approximately $0.55 \, \text{m}^2$.

Would you like further details or have any questions about the solution?

Here are 5 related questions you might find useful:

1. How is the scale factor used to relate volume and surface area in similar shapes?
2. Can we apply the same scale factor relationships to other types of shapes, like 2D shapes?
3. How would the problem change if we had a different ratio of volumes?
4. What real-life applications involve calculating surface areas and volumes of similar containers?
5. Why do volume and surface area scale differently when objects are similar?

Tip: When working with similar objects, remember that linear dimensions scale directly with the ratio, areas with the square of the ratio, and volumes with the cube.