00:01
So in this problem, we're told that a flare is getting fired from a boat, and the height in meters of the flare above the water at any time t seconds is represented by the function h of t, as you can see.
00:11
So the first thing we want to do is we want to figure out how long does it take the flare to reach its maximum height, and then to actually figure out what is the maximum height.
00:19
Well, as you can see, we have a quadratic function, and the maximum height is going to occur at the vertex.
00:26
So we can use our formula to find what typically would be our x value of our vertex to find after how long it will actually get to reach the maximum height.
00:34
Now, instead of finding the x value of our vertex, we're finding the t value because our equation is t in this case.
00:39
So the formula is t equals negative b over 2 times a.
00:43
Now, b represents or is the coefficient in front of t, so we're going to have negative 63, and a is the coefficient in front of our t squared term, so that would be negative 5 .25.
00:54
All right, so let's go ahead and simplify.
00:56
So i'm going to bring down negative 63, and then we have 2 times negative 5 .25, which is equal to negative 10 .5.
01:03
So to find the time, we'll take negative 63 and divide by negative 10 .5, which is equal to 6.
01:09
So it's going to take 6 seconds for the rocket to, or for the flare to reach its maximum height.
01:14
Now, the second question says, well, what will actually be the height? well, to find the height, we know the time it takes, we just need to substitute 6 in place of t in our function.
01:22
So in other words, we're looking for h of 6.
01:24
So this will equal to negative 5 .25 times 6 squared plus 63 times 6 plus 5.
01:32
So now we can go ahead and put this into our calculator.
01:35
So we're going to do negative 5 .25 times 6 squared plus 63 times 6 plus 5.
01:42
And you'll find that the maximum height of the flare will be 194 meters.
01:49
All right, so now let's take a look at the third question.
01:51
The third question says, how long does it take for the flare to hit the water? well, if the flare is hitting the water, that means the height would equal to zero.
01:59
So we're going to set h of t equal to zero and solve for t.
02:02
So negative 5 .25 times t squared plus 63t plus 5.
02:08
So to solve our quadratic equation, i'm going to use the quadratic formula...