Question Video: Writing an Expression for the Volume of a Cuboid Mathematics • First Year of Preparatory School

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.