A suspension bridge has two towers that rise 500 feet
above the road and are connected by cables that hang
in the shape of a parabola that can be represented by y
is equal to 1/8,960 times x minus 2,100 squared plus 8,
where x and y are measured in feet and x is the distance
from the left tower in the direction of the right tower.
What is the distance between the two towers?
Let's draw ourselves a diagram.
If we draw the leftmost tower-- I'm
going to do it in orange.
So we have the left tower right here.
That tower is at x is equal to 0.
We'll call that x is equal to 0, or maybe just call that the
0 point right there.
Then x is just measuring the distance from that leftmost
tower to the rightmost tower, so the rightmost tower in the
direction of the right tower.
This right here, maybe the right tower
is right over here.
They're both 500 feet high.
The right tower is right over there, and we don't
know where it is.
That's we have to figure out.
We have to figure out the distance between the two, but
we just need to know that x is measuring this distance.
It's measuring the distance from the left tower in the
direction of the right tower, so it's just traditional, just
like our coordinate plane.
That is what x is measuring.
If you say x is equal to a million feet, you're going to
get something way beyond that second tower.
If you say x is one foot, you're going to get something
in the direction of the second tower from the left tower.
So that's x, and then y, we can assume is the height of
the towers, or is just the height in general that we're
dealing with.
This is the y-axis, so we're just dealing in the first
quadrant of the coordinate plane, and that is y.
They're telling us that this parabola, the shape of the
parabola, can be represented-- or even better, they're saying
that the cable that connects these two is in
the shape of a parabola.
Let me draw that parabola right there.
They're the same height, so the parabola will look
something like this.
What we can do to figure out, or one possible technique--
let me draw a better parabola there.
That's even worse.
That's about as good as I can do.
One technique for finding the distance between the two is if
we could figure out the minimum point of this
parabola, which is also going to be the vertex of that
parabola, that's going to be halfway
between the two towers.
If we can figure out this x-coordinate, we can say
that's going to be halfway between the two towers and
then just multiply by two.
Let's see if we can figure out this minimum point, because
the parabola is symmetric around it, and these are both
at the same height.
Let's see if we can figure out that minimum point.
Let me rewrite this equation: We have y is equal to 1/8,960
times x minus 2,100 squared plus 8.
How do we find the minimum point here?
This whole expression that's squared, this whole expression
is always going to be positive, or it's actually
always going to be non-negative.
It could be equal to 0, and so this whole thing is always
going to be greater than or equal to 0.
This whole thing I'm squaring off in orange is always going
to be greater than or equal to 0.
So what is the lowest value that y can take on?
The lowest valued that y can take on will happen when this
expression is equal to 0, and this expression equals 0 only
when x minus 2,100 squared is equal to 0.
That's only going to be equal to 0 when x minus 2,100 is
equal to 0, so x-- let me write it this way-- equals 0,
and I'm just talking about this orange part-- when x
minus 2,100 is equal to 0, or if you add 2,100 to both
sides, or x is equal to 2,100.
When x is equal to 2,100, just out of curiosity, what is y?
If x is 2,100, this thing is 0 times 0, and y is just going
to be equal to that 8 there.
So this coordinate right here is the coordinate x is equal
to 2,100, y is equal to 8, and it's going to be halfway
between our two towers.
If this is at 2,100, how far is this tower away?
Well, it's going to be twice 2,100.
It's going to be 4,200, and everything we're
dealing with is feet.
It's going to be 4,200 feet away from that first tower.
So the distance between the two towers is 4,200 feet.