To find the volume of a rectangular prism, we use the formula:
[tex]\[ V = l \cdot w \cdot h \][/tex]
Given the dimensions of the rectangular prism:
- Length ([tex]\(l\)[/tex]) = [tex]\(5a\)[/tex]
- Width ([tex]\(w\)[/tex]) = [tex]\(2a\)[/tex]
- Height ([tex]\(h\)[/tex]) = [tex]\(a^3 - 3a^2 + a\)[/tex]
First, let's multiply the length and width:
[tex]\[ l \cdot w = (5a) \cdot (2a) = 10a^2 \][/tex]
Now, we need to multiply this result by the height [tex]\(h\)[/tex]:
[tex]\[ V = (10a^2) \cdot (a^3 - 3a^2 + a) \][/tex]
Expanding this product yields:
[tex]\[ V = 10a^2 \cdot a^3 - 10a^2 \cdot 3a^2 + 10a^2 \cdot a \][/tex]
Simplifying each term:
[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]
Combining these terms, we get the volume of the rectangular prism:
[tex]\[ V = 10a^5 - 30a^4 + 10a^3 \][/tex]
Therefore, the correct answer is:
[tex]\[ 10a^5 - 30a^4 + 10a^3 \][/tex]
So, the correct option from the given choices is:
[tex]\[ \boxed{10 a^5 - 30 a^4 + 10 a^3} \][/tex]
