Sure, let’s write a detailed, step-by-step solution to determine the total volume of the rectangular prisms and express it in a simplified form.
### Step-by-Step Solution
1. Identify Dimensions and Volumes of Each Prism:
- Red Prism (Bottom):
- Height: [tex]\(a\)[/tex]
- Length: [tex]\(b\)[/tex]
- Width: [tex]\(a - b\)[/tex]
- Volume: [tex]\(a \cdot b \cdot (a - b) = ab(a - b)\)[/tex]
- Blue Prism (Top Back):
- Height: [tex]\(b\)[/tex]
- Length: [tex]\(b\)[/tex]
- Width: [tex]\(a + b\)[/tex]
- Volume: [tex]\(b \cdot b \cdot (a + b) = b^2(a + b)\)[/tex]
- Green Prisms (Top Front and Top Small Piece):
- Height: [tex]\(a\)[/tex]
- Length: [tex]\(a\)[/tex]
- Width: [tex]\(a - b\)[/tex]
- Volume: [tex]\(a \cdot a \cdot (a - b) = a^2(a - b)\)[/tex]
2. Total Volume Calculation:
We sum the volumes of each individual prism to get the total volume of the combined structure.
[tex]\[
\text{Total Volume} = \text{Volume of Red Prism} + \text{Volume of Blue Prism} + \text{Volume of Green Prisms}
\][/tex]
Substitute the expressions for each volume:
[tex]\[
\text{Total Volume} = ab(a - b) + b^2(a + b) + a^2(a - b)
\][/tex]
3. Simplify the Expression:
Combine the terms and simplify the expression.
[tex]\[
\text{Total Volume} = ab(a - b) + b^2(a + b) + a^2(a - b)
\][/tex]
Distribute each term:
[tex]\[
ab(a - b) = a^2b - ab^2
\][/tex]
[tex]\[
b^2(a + b) = ab^2 + b^3
\][/tex]
[tex]\[
a^2(a - b) = a^3 - a^2b
\][/tex]
Combine all the terms:
[tex]\[
\text{Total Volume} = (a^2b - ab^2) + (ab^2 + b^3) + (a^3 - a^2b)
\][/tex]
Observe that some terms cancel each other out:
[tex]\[
a^2b - a^2b - ab^2 + ab^2 + b^3 + a^3
\][/tex]
This simplifies to:
[tex]\[
\text{Total Volume} = a^3 + b^3
\][/tex]
4. Express the simplified form:
Notice that [tex]\(a^3 + b^3\)[/tex] can also be factored as:
[tex]\[
a^3 + b^3 = (a + b)(a^2 - ab + b^2)
\][/tex]
Therefore, the simplified expression for the total volume of the combined prisms is:
[tex]\[
\boxed{(a + b)(a^2 - ab + b^2)}
\][/tex]
