00:01
So a tunnel with a parabolic arch is 12 meters wide.
00:04
If the height of the arch is 4 meters from the left edge is 6, then can we want to figure out if a truck that is 5 meters tall and 3 .5 meters wide can pass through the tunnel? so i'm going to draw a picture of my tunnel so that i can think about what it looks like.
00:18
It's a parabola, it makes a little arch here.
00:21
And we know that it's 12 meters wide, so if we were going to put a point here, that would be the point 6, 0, because half of 12 is 6.
00:29
I also know that up here, this is the vertex, which is my h, k.
00:34
And i know that this one has an x value of 0, because i put it on the y -axis.
00:39
And i don't know what the h value is, but we can find that later.
00:43
Then we also know that a point that it says 4 meters from the left edge or 4 meters from the right edge, it's going to give you the same answer because this is symmetric.
00:52
So 4 is 6 feet up, so that gives us a point on our parabola.
01:00
So essentially what we have is we have 3 points, and we can use these, or we have 2 points and the axis of symmetry.
01:06
And we can use this information to solve for the equation representing the parabola.
01:12
And then once we have that equation, we can use it to figure out whether the truck will fit.
01:18
So let's go ahead and get started.
01:20
We're going to actually create a system of equations.
01:22
We're going to be focusing, or using, the vertex form of the equation, which is y equals a times x minus h squared plus k.
01:31
H, k is your vertex, a is your coefficient in front, tells you your stretch or compression.
01:38
I'm going to plug in the numbers that i know.
01:40
I know that i have a point here at 6, 0, so y is 0, so i'm going to put a 0 here, equals a.
01:48
I don't know that, but i do know that x is 6, so i'm going to go ahead and put a 6 here.
01:55
H is 0 from up here, we can see that, minus 0 squared plus k.
02:02
So i don't know those other things, those are going to be my end notes.
02:05
So what i'm going to do my math, i end up with 0 equals 36a plus k.
02:13
So this is one equation that we're going to use, and now we need to find the other equation by substituting in different values.
02:22
So to find the other equation, we're kind of going to do the same thing, except we're going to use a different point.
02:26
Our y value is 6, so we're going to substitute a 6 for y, equals a times the x value.
02:33
The x value here is 4, and then times the h value, or minus the h value, 0.
02:42
All right, now we're going to simplify this one as well.
02:46
We're going to end up with 6 equals 16a plus k.
02:50
These are the two equations we're going to use to find our a and k, and then we'll be able to write the equation of the whole parabola...