Video Transcript
A rectangular prism has its base
area given by the trinomial two 𝑥 squared plus seven 𝑥𝑦 plus two 𝑦 squared
centimeters squared and its height given by the binomial 𝑥 plus 𝑦 centimeters. Find its volume.
In this question, we have been
asked to find the volume of a rectangular prism. So let’s start by recalling the
formula for the volume of a prism. We have that the volume of a prism
is equal to the area of its base multiplied by its height. We’ve been told in the question
that the area of the base of the prism is two 𝑥 squared plus seven 𝑥𝑦 plus two 𝑦
squared centimeters squared and that its height is 𝑥 plus 𝑦 centimeters.
We can see that the volume 𝑣 is
equal to the product of a binomial and a trinomial. We can use the distributive
property of multiplication over addition to distribute the binomial term over each
of the trinomial terms. We now have three products of
monomials and binomials, which we can distribute. And now we have completely expanded
the parentheses. We may notice that there are some
like terms, which we can combine.
Now that we have combined these
like terms, we cannot simplify the right-hand side anymore. Since the area is given in
centimeters squared and the height is given in centimeters, we have that the volume
will be in centimeters cubed. Hence, we can say that the volume
of the prism is two 𝑥 cubed plus nine 𝑥 squared 𝑦 plus nine 𝑥𝑦 squared plus two
𝑦 cubed centimeters cubed.
