To solve this, we’ll start by recalling the formula for the volume of a rectangular prism:

$\text{Volume} = \text{length} \times \text{width} \times \text{height}$

Given:

• Length $= 3a - b$ meters
• Width $= 8ab$ meters
• Height $= 5ab$ meters

The expression for the volume $V$ of the prism becomes:

$V = (3a - b) \times 8ab \times 5ab$

1. Multiply the width and height terms first: $8ab \times 5ab = 40a^2b^2$

2. Now multiply this result by the length term: $V = (3a - b) \times 40a^2b^2$

3. Expand this expression:

   • Distribute $40a^2b^2$ to both terms in $3a - b$: $V = 3a \cdot 40a^2b^2 - b \cdot 40a^2b^2$
   • Simplify each term: $V = 120a^3b^2 - 40a^2b^3$

So, the expanded expression for the volume is:

$V = 120a^3b^2 - 40a^2b^3$

Would you like further explanation on any step?

Additional Questions

1. How would the volume change if the height were doubled?
2. What happens to the volume if $b = 0$?
3. How could this formula be used to find the dimensions given a specific volume?
4. What if the length were given as $a + b$ instead of $3a - b$?
5. How would we proceed if we had values for $a$ and $b$ instead of expressions?

Tip

To verify expanded expressions, consider substituting sample values for variables and checking calculations at each step.