00:01
So in this problem, we're told that a bridge is built in the shape of a parabolic arch.
00:05
And we know the bridge has a span of 160 feet and it has a maximum height of 45 feet.
00:09
So the first thing we want to do is we want to find the right in the equation to represent this parabola.
00:15
So almost think we have this parabola here.
00:18
And we know that the span bit, meaning this distance right here, is equal to 160.
00:23
And we know that the height, which would hit the top of our parabola, would be 45 feet.
00:30
Well, if you think of what a parabola looks like, the maximum height is going to occur at the vertex.
00:35
So what we can do is i'm going to almost think like we're going to graph this on our coordinate plane, and i'm going to have the vertex be our origin.
00:43
So the vertex is going to be at the ordered pair, 0 -0.
00:46
So in this particular case, we know that our graph is heading downwards.
00:50
So if we use our vertex form of an equation for a quadratic, that would be y equals a times x minus h squared plus k.
00:59
Well, the vertex in this case would be the ordered pair hk.
01:03
So, in our case, because if we put this on our parabola when x is equal to 0, 0, that would mean our h and k values are 0.
01:10
So what we can do, we'll have y equals a times x minus 0 squared plus 0.
01:16
Well, x minus 0 is just x.
01:18
So really, we just have y equal to x squared.
01:21
So y equals a times x squared.
01:23
So we just have to find our value for a.
01:25
Well, remember that this distance is 45 feet down and this cutting the parabola in half.
01:30
So because the span was equal to 60, that would mean that this distance right here would be 80.
01:35
So we could represent this ordered pair.
01:37
It would be the order of paired 80, negative 45.
01:40
So now what we can do is we can substitute this in place of x and y in our function to find the value of a.
01:46
So we'll have negative 45 equal to a times 80 squared.
01:51
Well, first we have 80 squared, which is equal to 64.
01:55
So we're going to have negative 45 equal to a times 6 ,400.
02:02
So then we're going to divide both sides of our equation by 6 ,400.
02:07
So if we go ahead and do this, we're going to take negative 45 and we're divided by 6 ,400, which means if we were to reduce this fraction, that would end up with negative 9 over 1 ,280.
02:20
This would be our a value...