[MUSIC] Hello. I'm at the Knowlton School of
Architecture at the Ohio State University.
And today, we're going to measure the slope of some staircases, namely, this
staircase here and this even larger staircase.
You may think that this larger staircase is steeper than this staircase because,
well, this staircase is harder to climb. But let's actually find out.
And we're going to find out by measuring how far the staircase goes up by how wide
each step is. When I measure this step, I see that it
goes over twelve inches and it rises about six and a quarter inches.
On the other hand, we have this larger staircase here.
Let's see what its slope is. So, when I measure, I get 36 inches for
the width of each step. On the other hand, [SOUND] I find that
the height of each step is about 18-3/4 inches.
Now, that we've made the measurements, let's go compute the slope.
So, we've measured our stairs and we found out that the small steps went up by
6.25 inches and they were 12 inches wide, okay?
This gives us a slope for the small steps of 6.25 all over twelve.
But there were also large steps. Let's draw those in.
Here we go. So, this was one of the large steps and
then, we had another one. It goes off and it builds, it climbs over
there. It goes down like this.
This is a large step. Alright.
The height of the large step was this place right here.
18.75 inches, that's the height of the large step.
And the width of the large step was 36 inches.
Okay, great. So, the slope of the large steps is 18.75 all over 36.
These two slopes are the same, because we can simplify, we can reduce this fraction
to be dividing the numerator by three and dividing the denominator by three, we get
6.25 / 12. And we can see, if we connect the tips of
these stairs with a line or this makes a line of the desired slope and see how the
large stairs and the small stairs all meet at the corner of this line.
The fact that the staircase with the large steps and the staircase with the
small steps have exactly the same slope is evidenced by the fact that they have
the same railing. There's the railing for the staircase
with the large steps and here's the railing for the staircase with the small
steps. And they're exactly at the same angle.
So, we've seen that the staircase with the larger steps and the staircase with
the smaller steps actually have exactly the same slope.
Now, let's use the slope to estimate how tall the staircase is after you've gone a
certain distance in the horizontal direction.
How high up am I now? Here, you can see that we have a tape
measure where we're measuring the width or the horizontal distance that I
traveled here on the staircase. And if we get to the point where I was
standing, we'll find out that it's about 23 feet.
Now, how high was I standing when I was up there?
So, now that we've made our calculation, let's check our calculation.
Okay. Now, we have a question. We went to a certain part on the
staircase, say, right here.
And then we measured. We, we, we, we marked,
we had a point in mind and we measured the distance on the ground to that point.
We found out that it was 23 feet, okay?
So, if you remember, the slope of a staircase was 6.25 / 12, which is
approximately 0.52. We want to know this height, x.
So, we went x / 23 to be approximately 0.52.
Solve for x. Multiply both sides by 23 and we get x is
approximately 0.52 times 23. 0.52 times 23 is 11.96.
So, x is approximately 12 feet. Now, let's go check our answer.
So here we are, checking our measurement. [MUSIC] And I can see, it's almost
exactly twelve feet. [MUSIC]