00:02
In this question, we're given that an archer shoots an arrow straight upward with an initial velocity of 128 feet per second from a height of 6 feet.
00:12
And therefore, the function ht equals negative 16 t squared plus 128 t plus 6.
00:20
We'll use that information to find the maximum height reached by the arrow and also how long it takes for the arrow to reach the ground.
00:27
Now to get the maximum height of this arrow, we have to analyze the function as follows.
00:37
So the function, this is a quadratic function and the coefficient of the first term is negative 16, meaning it's going to be an inverted u.
00:50
So the maximum height is at the vertex.
00:53
And the vertex has the coordinates, let's say, the vertex has a...
00:59
As a coordinate, t, t, and h.
01:09
Okay, so what we want to do is to get the maximum height h.
01:14
But for us to get the maximum height h, we need to get the value of t.
01:22
Now to get the value of t at the vertex, we want to rewrite this so that it takes the form ht equals, a multiplied by t minus b squared plus c you know just to make it have a perfect square within it so let's begin the process so h t equals negative 16 t squared plus 128 t plus six so first thing is to factorize and take the common factor negative 16 as a factor divided within the negative 16 so t squared minus 8 t plus 6 over actually minus again because this is plus of minus 16 6 over 16 so 6 over 16 is 6 over 16 is 3 over 8 next we want to complete this square so we want to complete this square.
02:41
To complete the square we're looking for the constant term c and the constant term c is b over 2 squared.
02:52
And in this case the constant term c is negative 8 over 2 squared which is negative 4 squared and that's 16.
03:03
So we have negative 16 t squared minus 8 t plus 16.
03:14
That's a perfect square minus 3 over 8.
03:20
But take note we have added negative 16 and in other words this is extra.
03:32
This is additional from what we had before.
03:35
So what have we added? we have added negative 16 negative 16 times.
03:41
That means what we need to do after this is to reverse that by adding 16 times 16 to the expression.
03:51
And that means we are adding 256 to the entire expression...