SOLUTION: Fireworks Problem... t: elapsed time in seconds v0: initial upward velocity in feet per second h(t): altitude in feet Let {{{h(t)= -16t^2+v[0]t}}}, where v0 is positive. A.

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Question 31653: Fireworks Problem...
t: elapsed time in seconds
v0: initial upward velocity in feet per second
h(t): altitude in feet
Let , where v0 is positive.
A. Use the disciminant to show that there are always two elapsed times at which altitude is 0.
B. Use your answer to part A or an analysis of quadratic functions to find the maximum altitude of the rocket.
Thank you!
Answer by mbarugel(146) (Show Source):
You can put this solution on YOUR website!
Hello!
The discriminant of a quadratic equation of the form is

In order for the equation to have two roots (ie, two values of x for which the result of the function is zero), this discriminant must be positive. In the quadratic equation you provide, we have:
A = -16
B = v0
C = 0
Therefore, the discriminant is:

Since v0 is greater than zero, then the disciminant is positive; so we know that the equation has two solutions. So we know that there are two values for t that make the altitude equal to 0.
When the quadratic coefficient in a quadratic equation is negative (in this case, it's -16), then the maximum of the equation can be found at its vertex, whose formula is . So we know that at time , the maximum altitude is attained.
In order to find the maximum altitude (what we've found is the TIME at which the maximum altitude is attained) we simply plug the t we found into the quadratic equation. So the maximum altitude will be: