[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]