- [Instructor] We are told a rocket is launched
from a platform.
Its height in meters, x seconds after the launch is modeled
by h of x is equal to negative four times x plus two
times x minus 18.
Now, the first thing they ask us is what is the height
of the rocket at
the time of launch?
Pause the video and see if you can figure that out.
Well, what is x at the time of launch?
Well, x is the number of seconds after the launch, so at
the time of launch, x would be equal to zero.
The height of the rocket when x is equal to zero, they're
essentially saying, "Well, what is h of zero?"
To figure out h of zero, we just have to go back to this
expression and replace all the xs with zeros.
H of zero is going to be equal to negative four, negative
four times zero plus two, which is just going to be two
times zero minus 18, which is just going to be
negative 18.
Let's see, this is going to be negative eight times
negative 18, negative eight times negative 18,
which is the same thing as
negative eight times
negative nine
times two.
This is going to be positive 72 times two,
which is 144,
so 144
meters.
Did I do that right?
Let's see.
We're going to have ...
Yep that sounds right.
That's right.
How many seconds after launch will the rocket
hit the ground?
Pause this video again and see if you can answer that.
Well, what does it mean for the rocket to hit the ground?
That means that the height is equal to zero, so if you want
to figure out how many seconds after launch, how many
seconds that's x, so we want to figure out the x when our
height is equal to zero.
We can set up an equation.
Let's make our height h of x equal to zero, so zero is
equal to negative four times x plus two
times
x
minus 18.
Well, if you have the product of three different things
being equal to zero, the way you get this to be equal to
zero is if
at least one of these three things is equal to zero.
Well, negative four can't be equal to zero, so we could say
x plus two
equals zero.
I got that from right over here.
If x plus two were equaling to zero, then this equation
would be satisfied.
That would be the situation when x is equal to negative
two, but remember x is the number of seconds after the
launch, so a negative x would mean be going
before the launch.
We can rule that one out.
Then, we could also think about, "Well, x minus 18 if
"that's equal to zero, then this entire expression
"could be equal to zero."
X minus 18 equals zero, you add 18 to both sides.
You get x is equal to 18,
so 18 seconds after launch, well
we're going forward in time.
18
seconds after launch,
we see that our height is zero, we
have hit the ground.
Next question, how many seconds after being launched will
the rocket reach its maximum height?
Pause the video again and see if you can figure that out.
Well, the key realization here is
if you have a curve,
if you have a parabola in particular and it's going to look
something like this, if you're gonna have a parabola that
looks something like this, you're going to hit your maximum
height right over here between your two zeros
or between the two times that your height is zero.
If you figure out this x value and this x value, the
average of the two will give you your x value the time
after launch, when you're at your maximum height.
Well, we already figured out what this x-value is and what
this x-value is.
We know that h of x is equal to zero when x is either
equal to 18,
so that is x is equal to 18 or
x is equal to negative two,
so that is x is equal to negative two.
To answer this question, we just have to go halfway between
negative two and 18, so let's do that.
Negative two
plus
18
divided by two
gets us what?
That's going to be 16 over two, which is going to be equal
to eight.
This is right over here.
This is x equals eight seconds the rocket is at
its maximum height.
Last question, what is the maximum height that the rocket
will reach?
Once again, pause the video and try to answer that.
Well, we already know from the previous question that we
reach our maximum height when x is equal to eight,
eight seconds after launch.
To figure out the height then, we just have to evaluate
what h of eight is,
h of eight.
Remember that's what this function does, you give me any x
value, any elapsed time after launch and it will give me
the height, so eight seconds after launch, I know I have
maximum height.
To figure out that height, I just input it into the
function, so h of eight is going to be equal to negative
four
times
eight plus two times
eight minus
18.
Eight plus two is 10.
Eight minus 18
is negative
10.
You have negative four times
negative 100, so that's going to be positive 400.
H is given in meters, so that's its maximum height,
400 meters.