Certainly! Let's go through the process of deriving the expression for the volume of the gray-banded kingsnake enclosure given the provided conditions.
1. Identify the Variables:
- Let [tex]\( w \)[/tex] represent the width of the enclosure.
2. Express the Length in Terms of Width:
- According to the problem, the length ([tex]\( l \)[/tex]) is 20 inches more than the width.
- Thus, [tex]\( l = w + 20 \)[/tex].
3. Express the Height in Terms of Width:
- The height ([tex]\( h \)[/tex]) is 5 inches more than twice the width.
- Hence, [tex]\( h = 2w + 5 \)[/tex].
4. Volume Formula:
- The volume ([tex]\( V \)[/tex]) of a rectangular enclosure is given by:
[tex]\[
V = l \times w \times h
\][/tex]
5. Substitute the Expressions:
- Substitute the expressions for length and height into the volume formula:
[tex]\[
V = (w + 20) \times w \times (2w + 5)
\][/tex]
6. Combine the Terms:
- Multiply the terms out to express the volume in a standard polynomial form:
[tex]\[
V = w(w + 20)(2w + 5)
\][/tex]
Therefore, the expression that models the volume of the enclosure is:
[tex]\[
V = w(w + 20)(2w + 5)
\][/tex]
In conclusion, the expression that correctly models the volume of the gray-banded kingsnake enclosure is:
[tex]\[
V = w(w + 20)(2w + 5)
\][/tex]
This concise expression fits the given requirements and shows how the dimensions of the enclosure depend on its width.
