In this video, we're going to explore an application
of quadratic equations. When objects are thrown
or hit into the air, both their path and height
can be modeled with quadratic equations, and
this is because of gravity. Here, we’re
given that a baseball is hit and the height
of the baseball, h, measured in feet, after
t seconds can be modeled by the equation h
equals -16t² + 45t + 4. We're then asked
four different questions about this model.
The first question we’re asked is, “What
is the height of the baseball the instant
it is hit?” The second question we’re
asked is, “What is the height of the baseball
after one second?” The third question we’re
asked is, “When does the baseball hit the
ground?” And the last question we’re asked
is, “When is the height of the baseball
30 feet?” Before we get started, let's look
at a couple things here. So we have two units.
We have two variables and two units. So height,
which is measured in feet. And t is our other
variable. And t is measured in seconds. So
as we look at these four equations, it's important
to note what's being asked. The first two
start with, what is the height? So we are
to find a height in each of those first two
questions. The second two questions, however,
start with when. And when is referring to
time. So we need to find a specific time for
which, in the first one, the baseball hits
the ground. And in the second one, when the
baseball has a height of 30 feet. Okay. So
now that we've identified those, let's get
started with this first question. What is
the height of the baseball the instant it
is hit? So the instant it is hit is when t
equals 0. So we're going to replace t with
0 and find the value of h. So here we're going
to use t equals 0. And so h will then equal
-16(0²) + 45(0) + 4. And simplifying that,
-16 (0²) is simply 0. We're adding, then,
45 times 0, which is also 0. And then we're
adding 4. So we see here that h is 4. So when
t equals 0, h equals 4. And if we summarize
that in a sentence, we'll say the instant
the baseball is hit, the height is 4 feet.
So, “The instant the baseball is hit, its
height is 4 feet.” Okay. And so let's take
one second here to draw what we think is going
to be happening with this baseball then. So
we have t and we have h. And starting at 4
feet, and then it's going to come into the
air. And then it's going to come back down.
So that picture will be helpful as we work
through these next few questions. The second
question, what is the height of the baseball
after one second? So this is telling us, then,
that t equals 1, and we need to find h. So
we're going to replace t with 1 and calculate
h. So h will then be -16(1²) + 45(1) + 4.
And looking at that, we’ll then have -16
+ 45 + 4. And that will simplify to 33. So
we can then answer this question and state
that, “The height after one second 
is 33 feet.” And I noticed just a second
ago that I wrote t equals 1 at the end of
this question. That was just a note for myself
as to what I needed to do. So let's make sure
it doesn't look like that's the answer. t
equals 1. We need to find h. And then this
first one, I wrote t equals 0. We know that
t equals 0, and we needed to find h there,
as well. Just to make sure that doesn't look
like that's how the question should be answered.
Looking at this third one, we're asked, “When
does the baseball hit the ground?” Now,
when it hits the ground, we have to think
about what's true. When it hits the ground
and looking at our visual, then it's going
to be some amount of time. And it's going
to be when it hits the ground, the height
is going to be 0. So what we need here, is
we need h equal to 0. And then we're going
to solve that equation. So we can replace
h with how it's been defined, which is -16t²
+ 45t + 4. And we're going to solve for that
equals 0. So setting that up, we get -16t²
+ 45t + 4 equals 0. And from there, we can
now use the quadratic formula to solve for
t. So we'll have t equal--. And then we need
the opposite of b, which in this case will
be -45 plus or minus--. And then we write
a radical. And inside that square root, we're
going to put b², which is 45² - 4ac, which
will be -4(-16)(4). So I have -45 plus or
minus the square root. And then of the quantity
45² - 4 (-16)(4). And that's all over 2a,
which is 2(-16). Doing some of these calculations
with a calculator, I'll get that t equals--.
And then in that numerator, I'm going to have
-45 plus or minus--. And I'll do this step-by-step.
So I've calculated 45² on my calculator and
found it's 2025. And then 4(-16)(4) is -256.
So I have 2025 - (-256) inside that radical.
And then that whole expression is over -32,
taking 2(-16). Simplifying this one step further,
I'll find that t is then equal to--. And in
that numerator, I'm going to have -45 plus
or minus the square root of 2281. And that's
all over -32. At this point, everything is
as simplified as it can be. And so I'll use
a calculator to approximate each of these
proposed solutions. So the first one I have
is that t equals -45 plus the square root
of 2281 all over -32. And when I put that
into a calculator and approximate it, I get
-0.086, rounding that to three decimal places.
The second proposed solution I have-- so,
or t is equal to--. And then I have -45 minus
the square root of 2281 all over -32. Now,
this one is probably going to be the more
realistic one. I have a negative minus the
number in the numerator divided by negative.
This should be positive. And indeed when I
calculate that using a calculator, I find
that t is approximately 2.899. So the negative
time doesn't apply here. We weren't measuring
the height of the baseball before it was hit
with this formula, so we're only going to
use t approximately 2.899. So only that one
is applicable. It's the only applicable solution
is that second one. Now, we've found that
only applicable solution, let's summarize
answering this question, “When does the
baseball hit the ground?” “The ball hits
the ground after about 2.899 seconds.” Moving
on to this last question, we're asked, “When
is the height of the baseball 30 feet?”
And again, this is a when question. And so
we're going to find a time, and we're given
that the height is 30 feet. So we're going
to set h equal to 30. We can now write this
equation as -16t² + 45t + 4 equals 30. As
we know, that h is equal to -16t² + 45t +
4. To solve this equation, we'll set it equal
to 0, and then use the quadratic formula.
So subtracting 30 from each side, we'll obtain
-16t² + 45t - 26 equal to 0. So using the
quadratic formula, we'll find that t is then
equal to--. And on the right-hand side, we'll
use the opposite of b, which is -45 plus or
minus--. And then inside a radical, we're
going to put b², which is 45² - 4ac, which
will be - 4(-16)(-26). And that's all divided
by 2a, which is 2(-16). Simplifying this,
I'll use a calculator to simplify what's inside
that radical. And so we'll get -45 plus or
minus the square root of 2025 - 1664. And
that's all still divided by -32, when I take
2(-16). Simplifying this one step further,
I'll find that t is then equal to -45 plus
or minus the square root of 361 all divided
by -32. Now the square root of 361. 361 is
actually a perfect square, so this will simplify
then 2t equals -45 plus or minus 19. And that's
all over -32. Now there are two solutions
here, and I'll go through simplifying these
to find the exact solutions separately. So
the first one I have is that t is (-45 + 19)/-32.
Or for the second one, I have t equal to (-45
- 19)/-32. Simplifying that first one, -45
+ 19 is in the numerator, and that will be
-26. So I have -26/-32. That then simplifies
to t equals 13/16. For the second solution,
I had -45 - 19 in the numerator. And so that
will be -64. And then that denominator is
-32. So that will simplify to t equals 2.
So I now have two solutions: 13/16 of a second
and 2 seconds. And so I can summarize, then,
the answer to this question. The question
was, “When is the height of the baseball
30 feet?” So I can say the height of the
baseball is 30 feet… So the height of the
baseball is 30 feet 13/16 of a second. So
13/16 seconds after it was hit, and again
2 seconds after it was hit.
