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# Evaluating expressions with variables: cubes

Video transcript

The surface area of a
cube is equal to the sum of the areas of its six sides. Let's just visualize that. I like to visualize things. So if that's the cube,
we can see three sides. Three sides are facing us. But then if it was
transparent, we see that there are actually
six sides of a cube. So there's this one--
one, two, three in front-- and then one--
this is the bottom. This is in the back, and
this is also in the back. So you have three
sides of the cube. So I believe what
they're saying. The surface area of a
cube with side length x-- so if this is
x, if this is x, if this is x-- is given by
the expression 6x squared. That also makes sense. The area of each side is going
to be x times x is x squared, and there's six of them. So it's going to be 6x squared. Jolene has two
cube-shaped containers that she wants to paint. One cube has side length 2. So this is one cube
right over here. I'll do my best to draw it. So this right over
here has side length 2, so that's its dimensions. The other cube has
side length 1.5. So the other cube is going
to be a little bit smaller. It has side length 1.5. So it's 1.5 by 1.5 by 1.5. What is the total surface
area that she has to paint? Well, we know that the
surface area of each cube is going to be 6x
squared, where x is the dimensions of that cube. So the surface area of
this cube right over here is going to be 6. And now-- let me do it in
that color of that cube-- it's going to be
6 times x, where x is the dimension of the cube. And then the cube all
has the same dimensions, so its length, width, and
depth is all the same. So for this cube,
the surface area is going to be 6
times 2 squared. And then the surface
area of this cube is going to be 6
times 1.5 squared. And if we want the total
surface area she has to paint, it's going to be the
sum of the two cubes. So we're just going to
add these two things. And so if we were to compute
this first one right over here, this is going to be 6 times 4. This is 24. And this one right
over here, this is going to be a
little bit hairier. Let's see. 15 times 15 is 225. So 1.5 times 1.5 is 2.25. So 1.5 squared is 2.25. And 2.25 times 6-- so let
me just multiply that out. 2.25 times 6. Let's see. We're going to have
6 times 5 is 30. 6 times 2 is 12, plus 3 is 15. 6 times 2 is 12, plus 1 is 13. I have two numbers behind
the decimal-- 13.5. So it's going to be 13.5. And if I add these
two together, this is going to be equal to
the total surface area that she has got to paint, is
going to be 37.5 square-- well, I guess they're not
giving us the units. Well, 37.5 is going to be the
total area of square units of whatever the
units happen to be.