SOLUTION: The altitude of a rocket t seconds after launch is given by the function h(t)=-6t^2+32T+3 where h is the height in feet. how long after the launch will it take for the rock

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Question 1140923: The altitude of a rocket t seconds after launch is given by the function
h(t)=-6t^2+32T+3 where h is the height in feet.
how long after the launch will it take for the rocket to reach it's maximum height?


please and thank you
Found 2 solutions by MathLover1, rothauserc:
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!
h%28t%29=-6t%5E2%2B32t%2B3
here you have downward parabola, and its maximum is vertex (h,k)
so, rewrite your equation in vertex form
h%28t%29=a%28t-h%29%5E2%2Bk
h%28t%29=%28-6t%5E2%2B32t%29%2B3
h%28t%29=6%28t%5E2-%2816%2F3%29t%29%2B3
h%28t%29=6%28t%5E2-%2816%2F3%29t%2Bb%5E2%29-%28-6%29b%5E2%2B3........b=%2816%2F3%29%2F2=16%2F6=8%2F3
h%28t%29=6%28t-8%2F3%29%5E2%2B6%288%2F3%29%5E2%2B3
h%28t%29=6%28t-8%2F3%29%5E2%2B6%2864%2F9%29%2B3
h%28t%29=6%28t-8%2F3%29%5E2%2B128%2F3%2B3
h%28t%29=6%28t-8%2F3%29%5E2%2B128%2F3%2B9%2F3
h%28t%29=6%28t-8%2F3%29%5E2%2B137%2F3
vertex is at (8%2F3,137%2F3)
the rocket to reach it's maximum height in 8%2F3 or 2.7 seconds


Answer by rothauserc(4718) About Me  (Show Source):
You can put this solution on YOUR website!
h(t)= -6t^2 +32T +3
:
be consistent in the formula, I assume T = t, then
:
h(t)= -6t^2 +32t +3
:
the t^2 coefficient does not look right(should be -16 instead of -6) but I will use the formula as you have stated it
:
the x-axis is time and the y-axis is height
:
time when maximum height is reached is the x-coordinate of the parabola's vertex
:
Note the graph of this equation is a parabola that curves downward
:
time when maximum height is reached = -b/2a = -32/2(-6) = 32/12 = 8/3 = 2 and 2/3 seconds
:
Note general form of a quadratic is ax^2 +bx +c
: