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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.

Express the volume of the figure, V, as a function of one of its dimensions, x. Write the function's equation as a polynomial in standard form.

Learn how to express the volume of a figure as a polynomial function of one of its dimensions. This problem involves algebraic polynomial expansion and understanding the volume of a rectangular prism. Suitable for students in Grades 9-11.

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