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# Solving a scale drawing word problem

Video transcript

Sally is an
architect who creates a blueprint of a
rectangular dining room. The area of the
actual dining room is 1,600 times
larger than the area of the dining room
on the blueprint. The length of the dining room
on the blueprint is 3 inches. What is the length of the
actual dining room in feet? So there's a couple of really
interesting things going on here. They give us the dimensions
of the blueprint in inches. We want the actual
length in feet. And then they tell us that the
area of the actual dining room is 1,600 times larger. So they're not saying that
the scale of the blueprint is at 1/1600. It's going to be
something less than that, and let's think about what
that scale is going to be. Let's just think about
some different scales. Let's say that this
is my blueprint, and this is the actual
reality of the dining room that we're thinking about. And my blueprint
is let's just say 1 by 1, just for the
sake of argument. Now, if this was a 1 by 1 square
and we increased the dimensions by a factor of 2, so
it's a 2 by 2 square, what's the area going to be? Well, this area
is going to be 4. This area is 1, this area is 4. So you notice that if we
increase by a factor of 2, it increase our area
by a factor of 4. Or another way of saying, if we
increase each of our dimensions by a factor of 2,
we're going to increase our area by a factor of 4. If instead we increased
each of our dimensions by a factor of 3, this
would be a 3 by 3 square, and we would increase our
area by a factor of 9. So notice, whatever factor
we're increasing the area by, it's going to be the
factor that we're increasing the
dimensions by squared. So let's just think
about it that way. So they're telling us
that we're increasing the area by 1,600 times. Actually, let me just clean
this thing up a little bit. So one way we could imagine
it, if our drawing did have an area of 1,
which we can't assume, but we could for the
sake of just figuring out what the scale of
the drawing is. Let me clear all of this here. So the area of the actual dining
room is 1,600 times larger, and so if the drawing
had an area of 1, then the area of the
actual dining room would be 1,600 So
what would I have to multiply each of
the dimensions by to get an area factor of 1,600? Well, if I multiply
this dimension by 40 and this dimension by 40,
we see 40 times 40 is 1,600. You might say, hey, Sal,
how did you figure out 40? Well, the 16 is a big clue. We know that 4 times
4 is equal to 16, and so if you gave a 0 to each
of these 4's, if you made it 40 times 40, then that
is going to be 1,600. So this information
right over here tells us that the scale
factor of the lengths is 40. That would result in an scale
factor for the area of 1,600. So that's a good starting point. Now let's go to the actual
dining room on the blueprint. So the actual dining
room on the blueprint doesn't have these dimensions. We just used that to figure
out the scaling factor. The actual dining
room on the blueprint has a length of 3 inches. So maybe it looks
something like this. They don't give us any
of the other dimensions, so we can even imagine a 3 inch
by 2 inch, 1 inch, whatever we want. We could even imagine a
3 inch by 3 inch square. They only care about the length. Now let's multiply both of
these by a factor of 40. And we only care
about the length here. They actually say what's the
length of the actual dining room. So let's multiply it,
and obviously, this is not drawn to scale. Let's multiply this
times a factor of 40. So 3 times 40 is 120,
and this, of course, is what we're referring
to as the length. Now, you might be tempted
to say OK, we're done. This will be 120. But remember, this
is 120 inches. So what is 120 inches
in terms of feet? Well, 1 foot is
equal to 12 inches. If we were to multiply
both of these times 10, we know that 10 feet
is equal to 120 inches. Or another way you could
have thought about it, you have 120 inches divided
by 12 inches per foot is going to give you 10. So 120 divided by-- 120 inches--
let me write it this way. 120 inches divided
by 12 inches per foot is going to give you 10 feet. So that's the actual length
of the dining room in feet.