A rectangular prism has the following dimensions: l=5a

, w=2a
, and h= (a^3 -3a^3 +2)
. Use the formula v=l x w x h
to find the volume of the rectangular prism.

1 answer

To find the volume of the rectangular prism using the formula \( V = l \times w \times h \), we need to substitute the given dimensions into the formula.

The dimensions are:

  • Length (\( l \)) = \( 5a \)
  • Width (\( w \)) = \( 2a \)
  • Height (\( h \)) = \( a^3 - 3a^2 + 2 \)

First, we will simplify the expression for the height:

\[ h = a^3 - 3a^2 + 2 \]

Now we can calculate the volume:

\[ V = l \times w \times h = (5a) \times (2a) \times (a^3 - 3a^2 + 2) \]

Calculating \( (5a) \times (2a) \):

\[ 5a \times 2a = 10a^2 \]

Now substituting this back into the volume formula gives:

\[ V = 10a^2 \times (a^3 - 3a^2 + 2) \]

Next, we need to distribute \( 10a^2 \) across the polynomial \( (a^3 - 3a^2 + 2) \):

\[ V = 10a^2 \cdot a^3 - 10a^2 \cdot 3a^2 + 10a^2 \cdot 2 \]

Calculating each term:

  1. \( 10a^2 \cdot a^3 = 10a^{5} \)
  2. \( -10a^2 \cdot 3a^2 = -30a^{4} \)
  3. \( 10a^2 \cdot 2 = 20a^2 \)

Now we combine these results:

\[ V = 10a^{5} - 30a^{4} + 20a^{2} \]

So, the volume of the rectangular prism is:

\[ \boxed{10a^{5} - 30a^{4} + 20a^{2}} \]