To solve this problem, we need to use the properties of similar shapes. When two shapes are mathematically similar, the ratios of their corresponding dimensions are equal. Specifically:

1. The ratio of their volumes is the cube of their linear (scale) ratio.
2. The ratio of their surface areas is the square of their linear (scale) ratio.

Given:

• Volume of the larger container, $V_{\text{large}} = 3456 \, \text{cm}^3$
• Surface area of the larger container, $SA_{\text{large}} = 1024 \, \text{cm}^2$
• Volume of the smaller container, $V_{\text{small}} = 1458 \, \text{cm}^3$

We need to find the surface area of the smaller container, $SA_{\text{small}}$.

Step-by-Step Solution

1. Calculate the Volume Ratio:

   $\text{Volume Ratio} = \frac{V_{\text{small}}}{V_{\text{large}}} = \frac{1458}{3456}$

2. Determine the Linear Scale Factor:

   Since the volume ratio is the cube of the linear ratio $k$:

   $k^3 = \frac{1458}{3456}$

   Solving for $k$ (the linear scale factor):

   $k = \sqrt[3]{\frac{1458}{3456}}$

3. Calculate the Surface Area Ratio:

   The surface area ratio is $k^2$, so once we have $k$, we can calculate $k^2$ and use it to find the surface area of the smaller container:

   $SA_{\text{small}} = SA_{\text{large}} \times k^2$

Let's calculate these values step-by-step:

1. Calculate $\frac{1458}{3456}$.
2. Take the cube root to find $k$.
3. Square $k$ to get $k^2$.
4. Multiply $k^2$ by $1024 \, \text{cm}^2$ to find $SA_{\text{small}}$.

Let's proceed with these calculations.The surface area of the smaller container is $576 \, \text{cm}^2$.

Would you like further details on any part of this solution, or do you have additional questions?

Here are some related questions to explore further:

1. How would the solution change if only the surface area of the larger container were given?
2. What if we had the height of the containers instead of volume?
3. How can similar shape properties help in scaling down architectural models?
4. What happens to the surface area if the volume of the smaller container is halved?
5. How can we apply these principles in 3D printing?

Tip: For similar shapes, remember that area scales with the square of the linear ratio, and volume scales with the cube of the linear ratio.