Answered

A rectangular prism has the following dimensions: \(l=5a\), \(w=2a\), and \(h=\left(a^3-3a^2+a\right)\). Use the formula \(V=l \cdot w \cdot h\) to find the volume of the rectangular prism.

A. \(10a^6-30a^4+10a^2\)
B. \(10a^4-30a^3+10a^2\)
C. \(10a^5-3a^2+a\)
D. [tex]\(10a^5-30a^4+10a^3\)[/tex]



Answer :

GinnyAnswer
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.

Other Questions