To find the volume of a rectangular prism with given dimensions, we will use the volume formula \( V = l \cdot w \cdot h \), where:
- \( l = 5a \)
- \( w = 2a \)
- \( h = a^3 - 3a^2 + a \)
Let's substitute the given dimensions into the formula and calculate the volume step-by-step.
1. Substitute the values:
[tex]\[ V = (5a) \cdot (2a) \cdot (a^3 - 3a^2 + a) \][/tex]
2. Multiply the first two terms first:
[tex]\[ V = (5a \cdot 2a) \cdot (a^3 - 3a^2 + a) \][/tex]
[tex]\[ 5a \cdot 2a = 10a^2 \][/tex]
So,
[tex]\[ V = 10a^2 \cdot (a^3 - 3a^2 + a) \][/tex]
3. Distribute \( 10a^2 \) across each term inside the parentheses:
[tex]\[ V = 10a^2 \cdot a^3 - 10a^2 \cdot 3a^2 + 10a^2 \cdot a \][/tex]
[tex]\[ V = 10a^2 \cdot a^3 - 30a^4 + 10a^3 \][/tex]
4. Simplify each expression:
[tex]\[ 10a^2 \cdot a^3 = 10a^{2+3} = 10a^5 \][/tex]
[tex]\[ 10a^2 \cdot 3a^2 = 30a^{2+2} = 30a^4 \][/tex]
[tex]\[ 10a^2 \cdot a = 10a^{2+1} = 10a^3 \][/tex]
5. Combine all the simplified terms:
[tex]\[ V = 10a^5 - 30a^4 + 10a^3 \][/tex]
Therefore, the volume of the rectangular prism is:
[tex]\[ \boxed{10a^5 - 30a^4 + 10a^3} \][/tex]
This matches the fourth option provided in the question.
