Question

the movement of the progress bar may be uneven because questions can be worth more or less (including zero) depending on your answer. a rectangular prism has the following dimensions: $l = 5a$, $w = 2a$, and $h = (a^3 - 3a^2 + a)$. use the form find the volume of the rectangular prism. $\bigcirc\\ 10a^4 - 30a^3 + 10a^2$ $\bigcirc\\ 10a^5 - 3a^2 + a$ $\bigcirc\\ 10a^5 - 30a^4 + 10a^3$ $\bigcirc\\ 10a^6 - 30a^4 + 10a^2$

Explanation:

Step1: Recall the volume formula for a rectangular prism

The volume \( V \) of a rectangular prism is given by \( V = l \times w \times h \), where \( l \) is the length, \( w \) is the width, and \( h \) is the height.

Step2: Substitute the given values into the formula

We are given \( l = 5a \), \( w = 2a \), and \( h = a^{3}-3a^{2}+a \). So we substitute these into the volume formula:
\( V=(5a)\times(2a)\times(a^{3}-3a^{2}+a) \)

Step3: Multiply the first two terms

First, multiply \( 5a \) and \( 2a \). Using the rule of exponents \( x^{m}\times x^{n}=x^{m + n} \), we have \( (5a)\times(2a)=10a^{2} \)

Step4: Multiply the result by the third term

Now we multiply \( 10a^{2} \) with \( (a^{3}-3a^{2}+a) \). Using the distributive property \( c\times(d + e+f)=c\times d+c\times e + c\times f \), we get:
\( 10a^{2}\times a^{3}-10a^{2}\times3a^{2}+10a^{2}\times a \)

Step5: Simplify each term

For the first term: \( 10a^{2}\times a^{3}=10a^{2 + 3}=10a^{5} \)
For the second term: \( 10a^{2}\times3a^{2}=30a^{2+2}=30a^{4} \)
For the third term: \( 10a^{2}\times a = 10a^{2 + 1}=10a^{3} \)
Putting it all together, we have \( 10a^{5}-30a^{4}+10a^{3} \)

Answer:

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