00:00Hi everybody. So in this
00:02lesson we're going to do
00:02some quadratic modeling. Now the
00:06quadratic appear all over the
00:09place. Anything, these parabolas, anything
00:12where we have a parabola,
00:14we can model using quadratic.
00:17So look, any kind of
00:19projectile motion, like a basketball
00:20or tennis shot, fountains, U
00:26-shapes, like bridges,
00:28Arches look there's so many
00:30different places anywhere you just
00:33look around the room you'll
00:34probably be able to see
00:34some that looks like this
00:36or looks like this look
00:37out the window But this
00:40example what we're gonna look
00:41at is a distress flare
00:44fired into the air from
00:45a ship at sea and
00:46the height in meters of
00:47the flare above sea level
00:49is modeled by the quadratic
00:50function this Where T is
00:53the time in seconds after
00:54the flare was fired now
00:56just before, look this is
00:58gonna, this graph is gonna
00:59look like this. It's a
01:01negative quadratic. But just to
01:02be clear, the flare, they
01:06haven't fired the flare this
01:07way and it's going up
01:08and down like this. They're,
01:10they're, they've fired the flare
01:12straight up in the air.
01:14And it's just going up
01:14and down. And what's happening
01:16is because the x axis
01:18is time, this is time
01:20and height. As the time,
01:24so it's
01:24starts off here and then
01:27as the time increases it
01:29goes a bit higher, as
01:30the time increases a bit
01:31higher, a bit higher, a
01:33bit higher, but it's still,
01:35the flare is just going
01:36up, it's just going straight
01:38up and straight down again.
01:40And I just wanna make
01:41sure we're clear on that.
01:43Okay, right down the height
01:44from which the flare was
01:45fired, okay. Part A, it's
01:46right down, which means I
01:48don't have to show any
01:49working, the answer is 12.
01:51Why is it 12?
01:52Because the flare was fired
01:55at the beginning of this
01:57whole thing, which means when
01:59t was zero. If t
02:00is zero, that's zero, that's
02:02zero, and if it's 12.
02:03That's why it's 12 and
02:04it's 12 meters. Parpe. Find
02:07the height of the flare
02:0815 seconds after it was
02:10fired. Okay, I just need
02:11to sub in 15. It's
02:12h of 15 because t
02:14is 15. Number of seconds
02:16is 15, so it's negative
02:17zero point two times 15,
02:20squared plus 16 times 15
02:24plus 12. Do all that
02:27with the calculator. Now the
02:29way I'm gonna do it,
02:30I'm gonna show you how
02:31I'm gonna do it. I'm
02:32gonna draw this graph. So
02:35negative 0 .2, I'm gonna
02:38put x squared, remember your
02:40graph likes x not t.
02:44So plus 16x plus
02:4812 and we look something
02:49like this. Do I need
02:51to zoom out a bit?
02:55Maybe we could actually fit
02:57it put in our own
02:59window settings. So we could
03:01go from like negative 5
03:04to 100 and then let's
03:10go negative 10 to
03:16400. Looks okay. Okay. Find
03:27the height of the fair
03:2815 seconds after it was
03:29fair. So I basically want
03:30to find when x is
03:3115, what is y? Now
03:32the nice function to do,
03:34to do that is the
03:35trace function. So trace, graph
03:37trace, and then you just
03:39type in 15. And that
03:41gives you an x is
03:4115, what's
03:44y, 207. So h of
03:4715 is 207 meters. If
03:49you want to put that
03:50into your calculator, in fact,
03:52do just to kind of,
03:55well, just to make sure
03:57that it's the right answer.
03:59But put it put that
04:00into your normal calculator. So
04:02part C, the flare fell
04:04into the C, K seconds
04:06after it was fired to
04:07find the value of K.
04:08Let's go back to this.
04:10Now, when is it actually
04:12When is this falling? When
04:15does it go into the
04:16sea? Well, he starts... Why
04:19does it start at 12
04:20meters? Well, that's because he's
04:22at a ship. Remember, he's
04:26on a ship, which is
04:27probably 12 meters off the
04:29ground. So it starts at
04:3112. Now, where is the
04:33next? That's the Y intercept.
04:36Where's the X intercept? Well,
04:38there's two X intercepts. There's
04:39one over here, but this
04:40is
04:40like negative, negative time, that
04:43makes no sense. So this
04:44is where it's gonna fall
04:45into the C right here,
04:47because the X axis is
04:48the C. He starts off
04:50above the C there, and
04:52then it goes up, and
04:54it's gonna hit the C
04:55here. How do I find
04:56that? Well, with my analyze
04:58graph, zero, lower bound, upper
05:01bound, as 80 .74. So
05:05the working I'm gonna show
05:06is I wanna show that
05:07I understand that it's
05:08where the height, where the
05:14height equals zero. That's the
05:19solution. I don't have to
05:20do it. I don't have
05:20to do any more working.
05:22I've started up to use
05:22the quadratic formula or anything.
05:24I just write down t
05:26equals 80 .74. That's it.
05:3180 .74. Done.
05:36and actually it's k, so
05:40k equals 80 .74 because
05:47k is just, k is
05:49a time, it's k seconds
05:50after it was far. Okay,
05:52fine. C part D. Find
05:56the maximum height reached by
05:58the flare, so maximum height
06:03equals another
06:04I'm not even going to
06:05show any working here. It's
06:06just the maximum of this
06:08curve. So the maximum height
06:10is anelized graph, maximum lower
06:14bound, upper bound. So it's
06:16at 4332, which is the
06:18maximum height? Well, the x
06:19-axis is time. So the
06:21time is 40, and the
06:22height is 332. So the
06:24maximum height is 332 meters.
06:30Done. Easy.
06:33Part E, okay, I think
06:36this is the, right, this
06:38is the only tricky part
06:41of this question, I think.
06:43The nearest coast guard can
06:45see the flare when its
06:46height is more than 160
06:48meters above sea level, determine
06:50the total length of time
06:51the flare can be seen
06:53by the coast guard, okay,
06:56160 meters. Here is my,
06:59here is my,
07:01My quadratic. Where is 160
07:04meters, right? I'm going to
07:05draw the line 160. There,
07:12straight across. So look, the
07:15flare is going up and
07:16again, this is time, but
07:17whatever. The flare is going
07:18up, up, up, up. Now,
07:19once it's here, before it
07:21reaches the red line, the
07:23coast guard cannot see it.
07:26You cannot see the flare
07:27yet. Once it goes over
07:28the red line,
07:29Now the Coast Guard can
07:30see it. He can see
07:31it all the way from
07:32here to here and then
07:34it's gone down too far.
07:35You can't see it anymore.
07:36Maybe there's, I don't know,
07:38a mountain in the way
07:39or something. So you can't
07:40see it when it's below
07:42there. The question says find
07:43the total length of time.
07:45He can see it. So
07:46if I needed this time
07:47and this time, I could
07:48just subtract the two to
07:49find the total length of
07:50time. So let's find that.
07:52I need to do menu
07:53analyze graph intersection. I want
07:55to find the intersection of
07:56these two.
07:57So the lower bound and
07:58upper bound, 10 .67. And
08:03then I'm going to do,
08:05so that's 10 .67. And
08:07then I'm going to do
08:08analyze graph intersection, 69 .33.
08:15Now, before we actually use
08:16this values, let me, let
08:18me show you what working
08:18out I'm going to expect.
08:21So where does this minus
08:230 .2 t
08:25d squared plus 16t plus
08:2812. Where does this quadratic
08:30function equal 160? That's, that
08:33is me finding these two
08:36t's. These two times is
08:38where these two meet, when
08:40the height is 160. So
08:42the solutions are 10 .67,
08:45t equals 10 .67, comma,
08:50t equals 69 .33.
08:53The term in the total
08:58length of time, the flare
08:59can be seen. So he
09:00can see it after 10
09:02.67 and before 69 .33.
09:05So the total time, total
09:08time equals this minus this
09:1669 .33 minus 10 .67.
09:210 .33 minus 10 .67
09:27gives me 58 .66 equals
09:3058 .66 seconds. Okay, so
09:38they usually ask kind of
09:39one tricky question like that
09:42where you actually have to
09:43think about what's going on.
09:44But I feel once you
09:46can draw the quadratic with
09:48the graph of the
09:49quadratic function on your calculator,
09:51there's no reason why you
09:53can't find these ones. They're
09:55always looking for x intercepts,
09:58they're always looking for maximums
09:59or minimums, and they're always
10:02looking for a particular time
10:06at a height or a
10:07particular height at a time.
10:09And hopefully now you can
10:11do all those things.