Question

The design for a rectangular solid shipping container has a volume of 24x^3+6x^2-9x

Answer 1

To work with the given volume of the rectangular solid shipping container \( V(x) = 24x^3 + 6x^2 - 9x \), we can perform several operations such as factoring or analyzing the expression to understand more about its dimensions.

### Step 1: Factor the volume expression
We can start by factoring the polynomial \( V(x) \).

1. **Factor out the greatest common factor (GCF)**:
The GCF of the terms \( 24x^3, 6x^2, \) and \( -9x \) is \( 3x \). So, we factor out \( 3x \):

\[
V(x) = 3x(8x^2 + 2x - 3)
\]

2. **Factor the quadratic \( 8x^2 + 2x - 3 \)**:
We can factor the quadratic trinomial \( 8x^2 + 2x - 3 \) further. We look for two numbers that multiply to \( 8 \times -3 = -24 \) and add up to \( 2 \).

The pairs of factors of \( -24 \) that add to \( 2 \) are \( 6 \) and \( -4 \).

We can re-write the quadratic:
\[
8x^2 + 6x - 4x - 3
\]

Group the terms:
\[
= (8x^2 + 6x) + (-4x - 3)
\]
\[
= 2x(4x + 3) - 1(4x + 3)
\]
\[
= (2x - 1)(4x + 3)
\]

### Final Factorization
Putting all of this together, we can express \( V(x) \) as:
\[
V(x) = 3x(2x - 1)(4x + 3)
\]

### Step 2: Interpret the dimensions
Since \( V(x) \) represents the volume of a rectangular solid, we can interpret \( 3x, (2x - 1), \) and \( (4x + 3) \) as the three dimensions \( l, w, h \) of the rectangular solid:

- One dimension could be \( 3x \)
- The second could be \( 2x - 1 \)
- The third could be \( 4x + 3 \)

### Step 3: Conditions for dimensions to be valid
For physical dimensions, the expressions for width and height must be positive:

1. **For \( 3x > 0 \)**: \( x > 0 \)
2. **For \( 2x - 1 > 0 \)**: \( x > \frac{1}{2} \)
3. **For \( 4x + 3 > 0 \)**: This is always true for \( x > 0 \) since \( 4x + 3 \) will always be positive.

### Conclusion
The conditions determine that \( x \) needs to be greater than \( \frac{1}{2} \) for all dimensions to remain positive. The volume is factored as \( V(x) = 3x(2x - 1)(4x + 3) \), which gives us insight into the dimensions of the shipping container.