The problem asks to find the volume $V$ of a larger rectangular solid, given that two solids are similar. Here are the steps to solve the problem:

Step 1: Understand the Ratio of Side Lengths

Since the solids are similar, the ratio of their corresponding sides is constant. The side length of the smaller solid is $5 \, \text{m}$, and its volume is $125 \, \text{m}^3$. The side length of the larger solid is $6 \, \text{m}$.

Let the ratio of the side lengths be: $\text{Ratio} = \frac{\text{Side length of larger solid}}{\text{Side length of smaller solid}} = \frac{6 \, \text{m}}{5 \, \text{m}} = \frac{6}{5}$

Step 2: Cube the Ratio for Volume

The volumes of similar solids are proportional to the cube of the ratio of their corresponding side lengths. Therefore, the ratio of the volumes is: $\text{Volume ratio} = \left(\frac{6}{5}\right)^3$

Step 3: Calculate the Volume of the Larger Solid

Now, calculate the volume of the larger solid using the volume ratio: $V = \text{Volume of smaller solid} \times \left(\frac{6}{5}\right)^3$ Substituting the known values: $V = 125 \, \text{m}^3 \times \left(\frac{6}{5}\right)^3$

Let's calculate this.The volume $V$ of the larger solid is approximately $216 \, \text{m}^3$.

Would you like any further details on the calculation, or do you have any questions?

Here are five related questions for further practice:

1. What would the volume be if the side length of the larger solid were 7 meters instead of 6 meters?
2. How does the volume change if the side length of the smaller solid is reduced to 4 meters?
3. If the larger solid's volume is given as 343 cubic meters, what is its side length?
4. What is the ratio of surface areas between the two similar solids?
5. How would the calculation change if the solids were cylinders instead of rectangular solids?

Tip: Always remember that for similar solids, the volume ratio is the cube of the ratio of their corresponding side lengths.