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

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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) About Me  (Show Source):
You can put this solution on YOUR website!
a)
To find the time at the highest point, lets complete the square for H%28t%29=-16t%5E2%2B96t%2B4+ so we can convert it into vertex form:
Solved by pluggable solver: Completing the Square to Get a Quadratic into Vertex Form


y=-16+x%5E2%2B96+x%2B4 Start with the given equation



y-4=-16+x%5E2%2B96+x Subtract 4 from both sides



y-4=-16%28x%5E2-6x%29 Factor out the leading coefficient -16



Take half of the x coefficient -6 to get -3 (ie %281%2F2%29%28-6%29=-3).


Now square -3 to get 9 (ie %28-3%29%5E2=%28-3%29%28-3%29=9)





y-4=-16%28x%5E2-6x%2B9-9%29 Now add and subtract this value inside the parenthesis. Doing both the addition and subtraction of 9 does not change the equation




y-4=-16%28%28x-3%29%5E2-9%29 Now factor x%5E2-6x%2B9 to get %28x-3%29%5E2



y-4=-16%28x-3%29%5E2%2B16%289%29 Distribute



y-4=-16%28x-3%29%5E2%2B144 Multiply



y=-16%28x-3%29%5E2%2B144%2B4 Now add 4 to both sides to isolate y



y=-16%28x-3%29%5E2%2B148 Combine like terms




Now the quadratic is in vertex form y=a%28x-h%29%5E2%2Bk where a=-16, h=3, and k=148. Remember (h,k) is the vertex and "a" is the stretch/compression factor.




Check:


Notice if we graph the original equation y=-16x%5E2%2B96x%2B4 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-16x%5E2%2B96x%2B4%29 Graph of y=-16x%5E2%2B96x%2B4. Notice how the vertex is (3,148).



Notice if we graph the final equation y=-16%28x-3%29%5E2%2B148 we get:


graph%28500%2C500%2C-10%2C10%2C-10%2C10%2C-16%28x-3%29%5E2%2B148%29 Graph of y=-16%28x-3%29%5E2%2B148. Notice how the vertex is also (3,148).



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).