he​ volume, of a rectangular solid with length​ L, width​ W, and height H is given by the formula 

Upper V equals LWH.V=LWH. 

Use this formula to write a polynomial in standard form that​ models, or​ represents, the volume of the open box.

x

10-2x

The volume of the open box is 

​(Simplify your answer. Type your answer in standard​ form.)

To model the volume of the open box using the given formula \( V = LWH \), we'll need to represent the length \( L \), width \( W \), and height \( H \) of the box in terms of the variable \( x \).

Given:
- Height \( H = x \)
- Length \( L = 10 - 2x \)

We can infer that the width \( W \) would also be \( 10 - 2x \), assuming a square base with the same dimensions as the length.

The volume \( V \) of the box is given by:
\[
V = L \times W \times H
\]
Substituting the values:
\[
V = (10 - 2x) \times (10 - 2x) \times x
\]

First, expand \( (10 - 2x)(10 - 2x) \) using the distributive property (or FOIL method):
\[
(10 - 2x)(10 - 2x) = 10 \times 10 - 10 \times 2x - 2x \times 10 + (-2x) \times (-2x)
\]
\[
= 100 - 20x - 20x + 4x^2
\]
\[
= 100 - 40x + 4x^2
\]

Next, multiply this result by \( x \):
\[
V = (100 - 40x + 4x^2) \times x
\]
\[
= 100x - 40x^2 + 4x^3
\]

Finally, write the polynomial in standard form (arranging terms in descending order of degree):
\[
V = 4x^3 - 40x^2 + 100x
\]

So, the polynomial that models the volume of the open box is:
\[
\boxed{4x^3 - 40x^2 + 100x}
\]

Do you have any questions, or would you like more details?

Here are 5 related questions:
1. What happens to the volume when \( x \) increases beyond a certain point?
2. How would the formula change if the width \( W \) were different from the length \( L \)?
3. What are the dimensions of the box when \( x = 2 \)?
4. What is the volume of the box when \( x = 1 \)?
5. How do you find the maximum volume of the box given the polynomial?

**Tip:** Always expand binomials first before multiplying by other terms when finding the volume of solids like this.

Explore how to model the volume of an open box using a polynomial representation. Learn step-by-step how to derive the polynomial 4x^3 - 40x^2 + 100x from given dimensions. Suitable for high school students studying algebra and polynomial functions.

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