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This is going to be the second video about word problems that are solved with quadratic
equations,
and what I want to do this time
is something called a projectile problem.
Now projectiles
are simply things that are thrown or hit
or launched into the air.
So a baseball is a projectile,
a basketball,
a rock that's thrown in the air, a rocket.
Anything like that is a projectile.
So here's the problem:
A projectile is launched from a tower into the air with an initial
velocity of 48 feet per second.
Its height,
h, in feet
above the ground is modeled by the function
h = -16t-squared
+ v-sub-0 t
+ v-sub-0 t
plus 64 where t is the time in seconds since the projectile was
launched
and v-sub-0 is the initial velocity.
And then it asks three questions:
How long was the projectile in the air?
When did it reach its maximum height?
What was its maximum height?
Okay, I know this looks kind of scary,
but it doesn't have to be. It's just a lot of words
and we're given a lot of good information. We're given this formula.
So let's look at the formula.
h. We're told that h is height -- if something is flying through the air,
it's going to have some height.
So this formula is going to tell us what the height is
in terms of t, because we have h equals -16t-squared
and we're told that t is time.
Well that makes sense because
as time goes by is goes up than then it comes down again.
And then it has this v-sub-0.
And it says v-sub-0 is the initial velocity.
Well, we're told what the initial velocity is.
Initial velocity is 48 feet per second,
so let's rewrite this
with the 48.
Okay?
So we're going to have
h
equals -16
t-squared
plus... instead of the v-sub-0,
I'll have 48t
I'll have 48t
+ 64.
And then the question asked was
how long was it in the air?
Well it was in the air until it hit the ground,
and when it hit the ground its height was zero.
So I'm gonna take this original equation
and I'm just going to set it equal to zero.
So what do we have here? It's just a quadratic equation that we can solve.
Looking at it, we've got these big numbers .. 16, 48 64... so you
might want to try to the quadratic formula.
But let's take a look at the numbers.
They're all even,
so I could
divide out a 2 from his whole thing.
In fact, I could divide out a 4. All these numbers are divisible by 4.
And if we go even further, we're gonna see that we could
divide a 16 into each of these numbers.
So if we do that we get a very simple equation.
Let's divide, actually, a negative 16. That will get this negative sign
out of the way too.
If I divide all of this by -16,
I'm going to have
just t-squared
plus -16 into 48...
that's not a plus anymore, that's negative 3
t
and -16 into 64
is -4.
So now what I is this -16, which I don't have to worry about.
I have to keep writing it, but I don't have to worry about it...
times t-squared
minus 3t - 4
equals 0.
So let's factor this quadratic
equation that we have here.
So I have -16
and then
I'm gonna have
a t, and a t,
And I've got a -3 in the middle and a -4 at the end,
so I know I want one positive number and one negative number,
and
I can make a 4
with a 1 and a 4... 1 times 4 is 4
in the difference between 1 and 4 is 3. So those will be good numbers.
So let's put a 1 here and a 4 here.
I'm gonna make the 1 positive and 4 negative. That will give me -3t
in the middle.
Now,
I don't have to worry about the -16 anymore because I know
that that doesn't equal zero.
So either t + 1 equals 0 or t - 4 equals zero,
or both of them equal zero.
So let's say t plus 1 equals zero
and t - 4
equals zero.
So now that means t equals
-1
or
t equals
4.
Well,
I don't want a negative time,
so this -1 is not going to work.
So it seems like
the projectile is going to hit the ground
after 4 seconds. So that's the answer to the first part of the problem.
Let's go back and see what the other two questions were.
When did it reach its maximum height?
and what was its maximum height?
So let's erase this and just have
the basic equation that we had.
h equals
-16
t-squared
+ 48t
+ 64.
we want to know
when it reached its maximum height.
Well, this thing was a parabola.
It went up
and he came down.
So if want to find its highest point, we're basically looking for its vertex.
That's when it hits
its maximum height, its highest point.
So let's see... what's the vertex
of this quadratic equation?
Well, the vertex is going to be... remember the vertex formula is -b
over
2a.
Well, b is 48 so -b is -48.
a is -16. 2 times -16 is -32.
And I can reduce this whole thing
3/2.
or
1.5. So after one 1.5 seconds
the projectile was at its maximum height.
It also asked me what the maximum height was. Well the maximum height is going
to be how high it was
after 1.5 seconds
so let's put 1.5 in wherever
I've got a 't'.
So the height is going to equal -16
times
1.5 squared
plus
48
times
1.5
plus 64.
Let's see what that comes to.
This is going to be -16
times... 1.5 times 1.5
is 2.25,
is 2.25,
because 15 squared is 225...
and 48
times 1.5, or one and a half times 48...
so let's see... half of 48 is 24,
48 and 24 is 72...
plus
64.
And now I need
2.25 times -16.
Well let's see what I can do.
I know that's going to be negative. 2 times 16 is 32
and one quarter, .25 times 16 is 4,
so 32 and 4 is 36. That's -36
plus 72
plus 64
And what does that equal.
Well, 36 from 72 is 36.
And 36 plus 64
is 100.
So its maximum height is going to be
100.
And of course if you wanted to
you could take this information, take this equation
and put it into your calculator,
and you would also find out when it hit the ground.
That would be one of the zeros. That would be the point your calculator
that looks like that,
and you could find what the vertex is by using the maximum function in your calculator.
And that would give you the 100 for the height.
It would give you 1.5 for the seconds and so on. I just wanted to show you how
to do it mathematically.
So that's about it for now.
Take care,
I'll see you next time.