To express the volume of the figure $V$ as a function of one of its dimensions $x$, we can follow these steps:

1. Identify the dimensions of the rectangular prism:

The figure's dimensions are given as:

• Length: $6x - 1$
• Width: $x + 1$
• Height: $x + 8$

2. Write the volume function:

The volume $V(x)$ of a rectangular prism is calculated by multiplying its length, width, and height. Thus, we have: $V(x) = \text{Length} \times \text{Width} \times \text{Height}$ Substituting the given dimensions: $V(x) = (6x - 1) \times (x + 1) \times (x + 8)$

3. Expand the expression:

First, expand the product of the first two binomials: $(6x - 1)(x + 1) = 6x^2 + 6x - x - 1 = 6x^2 + 5x - 1$

Next, multiply this result by the third binomial $(x + 8)$: $V(x) = (6x^2 + 5x - 1)(x + 8)$

Distribute each term: $V(x) = 6x^2(x + 8) + 5x(x + 8) - 1(x + 8)$ $V(x) = 6x^3 + 48x^2 + 5x^2 + 40x - x - 8$

4. Simplify the expression:

Combine like terms: $V(x) = 6x^3 + (48x^2 + 5x^2) + (40x - x) - 8$ $V(x) = 6x^3 + 53x^2 + 39x - 8$

Thus, the simplified volume function is: $V(x) = 6x^3 + 53x^2 + 39x - 8$

This is the correct polynomial expression for the volume in standard form.

Do you want details or have any questions?

Here are five related questions:

1. How can you verify the expanded polynomial is correct?
2. What would be the volume if $x = 2$?
3. What happens to the volume as $x$ approaches 0?
4. How does the leading term $6x^3$ affect the graph of the volume function?
5. How can the volume function be used to find the dimensions of the figure for a specific volume?

Tip: Always check your work step-by-step, especially when expanding and combining like terms in polynomials.