Video Transcript
A rectangular prism has a square
base of length three 𝑥 plus two 𝑦 meters and a height of length two 𝑦 plus 𝑥
meters. Find its volume in terms of 𝑥 and
𝑦.
The volume of a rectangular prism
is equal to its height times the area of its base. Let’s work out the area of the
base. Since the base is square, this is
equal to its side length squared. We can do this multiplication using
a grid. We need to multiply the
corresponding terms. Three 𝑥 times three 𝑥 is nine 𝑥
squared. Three 𝑥 times two 𝑦 is six
𝑥𝑦. Two 𝑦 times three 𝑥 is six
𝑥𝑦. And two 𝑦 times two 𝑦 is four 𝑦
squared. We add all of this together and
collect the two middle terms, forming the trinomial nine 𝑥 squared plus 12𝑥𝑦 plus
four 𝑦 squared.
Multiplying this expression by the
height of two 𝑦 plus 𝑥 will give us an expression for the volume. And we can do this multiplication
using another grid. This yields 18𝑥 squared 𝑦 plus
24𝑥𝑦 squared plus eight 𝑦 cubed plus nine 𝑥 cubed plus 12𝑥 squared 𝑦 plus four
𝑥𝑦 squared. Collecting like terms, we have nine
𝑥 cubed plus 30𝑥 squared 𝑦 plus 28𝑥𝑦 squared plus eight 𝑦 cubed meters
cubed.
