Question

a rectangular prism has the following dimensions: ( l = 5a ), ( w = 2a ), and ( h = (a^3 - 3a^2 + a) ). use the formula ( v = l cdot w cdot h ) to find the volume of the rectangular prism.
options:
( 10a^5 - 30a^4 + 10a^3 )
( 10a^4 - 30a^3 + 10a^2 )
( 10a^6 - 30a^4 + 10a^2 )
( 10a^5 - 3a^2 + a )

Explanation:

Step1: Recall the volume formula

The volume \( V \) of a rectangular prism is given by \( V = l \cdot w \cdot h \). We know \( l = 5a \), \( w = 2a \), and \( h = a^{3}-3a^{2}+a \). First, multiply \( l \) and \( w \): \( 5a \cdot 2a = 10a^{2} \).

Step2: Multiply by height

Now, multiply the result by \( h \): \( 10a^{2} \cdot (a^{3}-3a^{2}+a) \). Using the distributive property (multiply \( 10a^{2} \) with each term inside the parentheses):

• \( 10a^{2} \cdot a^{3}=10a^{2 + 3}=10a^{5} \)
• \( 10a^{2} \cdot (-3a^{2})=-30a^{2+2}=-30a^{4} \)
• \( 10a^{2} \cdot a = 10a^{2 + 1}=10a^{3} \)

Combine these terms: \( 10a^{5}-30a^{4}+10a^{3} \).

Answer:

\( 10a^{5}-30a^{4}+10a^{3} \) (corresponding to the first option among the choices)