00:01
In this problem, our job is to determine how far from the wall should the catapult be placed.
00:07
So from previous work, we know the equation of this parabola.
00:11
Y is approximately negative 0 .0465x times x minus 59 .75.
00:18
So let's use that equation to draw a picture of the path of the projectile.
00:26
And we're going to place the catapult at 0 .0.
00:30
And we know that the vertex occurs when y is 41 .5, when x is 29 .875, and we have the other x intercept here at 59 .75.
00:58
So we know that the projectile will follow this path, this parabola.
01:05
And so we are trying to clear a castle wall, which will place right here.
01:18
It's 20 units in height.
01:22
We want to make sure that path clears that wall on the way down, but we don't want the projectile to hit the queen who is standing back here.
01:36
So we're looking for this value of x to make sure that the projectile clears the wall, but we have to avoid the queen who is standing back here at x plus 25 units.
01:49
So here's the basic picture of what we've got, what we're trying to accomplish.
01:55
So right here, if we're cutting it close, we could find the value of x where y is 20, but we might not want to cut it that close.
02:08
Maybe we should make sure we don't go below 21.
02:12
We can find that corresponding x.
02:15
We'll find both of those.
02:16
And notice there are two places where that happens on the way up and on the way down.
02:20
We want the value of x on the way down.
02:22
So let's take our equation and put in both 20 and 21 and use the quadratic formula to solve for x.
02:31
We'll come down here, just as a reminder, here's the equation we're using.
02:39
This gives the equation of the parabola.
02:45
So let's see what happens when y is 20.
02:53
We have this equation.
02:59
We need to solve for x...