Video Transcript
Find the area of the shaded
region.
So essentially, we’ll want the area
of the two pink triangles. And the area of a triangle is
one-half times the base times the height. Now we know the height of the
triangles. It’s six 𝑥. However, we know the basis
completely together to be 15𝑥 plus five. So let’s maybe find a different
way, because the 15𝑥 plus five we’re not sure how to split them up.
Instead, let’s think of this as an
entire rectangle area and then taking away that white triangle, this one. And this will give us the area of
the shaded region. The area of a rectangle is length
times width, and the area of a triangle is one-half times the base times the
height. So we know the length and the width
of the rectangle. So we need to take 15𝑥 plus five
times six 𝑥.
And then the triangle, if we will
look at this upside down, it might be a little easier. But we will use this as the base,
the 15𝑥 plus five, and the height of that triangle will be six 𝑥. So one-half base times height would
be one-half times 15𝑥 plus five times six 𝑥.
Let’s go ahead and rewrite it down
here so we have more room, because we need to distribute. 15𝑥 times six 𝑥 is 90𝑥 squared,
and five times six 𝑥 would be 30𝑥. Bring down our subtraction
sign. We can go ahead and take one-half
times six, which is three. So we can think of this is as 15𝑥
plus five times three 𝑥. 15𝑥 times three 𝑥 is 45𝑥
squared.
Now it is important to put a
bracket out front. That way, we recognise the minus
sign — that’s pink — will need to be distributed to everything in the blue. And five times three 𝑥 would be
15𝑥. So now we need to distribute this
negative sign. So we actually have minus 45𝑥
squared minus 15𝑥.
Now we can combine like terms. 90𝑥 squared minus 45𝑥 squared
would be 45𝑥 squared. 30𝑥 minus 15𝑥 would be 15𝑥. Therefore, 45𝑥 squared plus 15𝑥
will be the area of the shaded region. It may also be written as 15𝑥
times three 𝑥 plus one, because we could take out a greatest common factor of 15𝑥
from both terms. Either answer will probably be
fine.
