SOLUTION: For independence Day, the village of treetop is planning to launch fireworks from a barge on the lake. The rockets will achieve a height represented by the function H(t)=-16t^2+96t

Question 77426: For independence Day, the village of treetop is planning to launch fireworks from a barge on the lake. The rockets will achieve a height represented by the function H(t)=-16t^2+96t+4 where h(t) is measured in feet and t is time in seconds after launch.
A) At what time to the nearest second is the fireworks rocket at its highest point ?
B) if the rocket is designed to explode at its highest point, to the nearest foot, how high is the rocket when it explodes?
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
a)
To find the time at the highest point, lets complete the square for so we can convert it into vertex form:

Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form

Start with the given equation

Subtract from both sides

Factor out the leading coefficient

Take half of the x coefficient to get (ie ).

Now square to get (ie )

Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of does not change the equation

Now factor to get

Distribute

Multiply

Now add to both sides to isolate y

Combine like terms

Now the quadratic is in vertex form where , , and . Remember (h,k) is the vertex and "a" is the stretch/compression factor.

Check:

Notice if we graph the original equation we get:

Graph of . Notice how the vertex is (,).

Notice if we graph the final equation we get:

Graph of . Notice how the vertex is also (,).

So if these two equations were graphed on the same coordinate plane, one would overlap another perfectly. So this visually verifies our answer.

So the vertex is (3,148) where the first coordinate is t, so t=3.

b)
Since the rocket explodes at the vertex (the highest point), the height of the explosion is 148 ft (ie it's the y coordinate of the vertex).
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