The airplane reaches its maximum height of 81 feet in 12 seconds.
If y = [tex]x^{2}[/tex], it means that the second derivate is 2 which is positive, then there is a minimum turning point.
If y = - [tex]x^{2}[/tex], it means that the second derivative will be -2 which is negative, then there is a maximum turning point.
Analysis:
h(t) = -[tex](t - 12)^{2}[/tex] + 81
By expanding,
h(t) = -([tex]t^{2}[/tex] - 24t + 144) +81
h(t) = -[tex]t^{2}[/tex] + 24t - 144 + 81
h(t) = -[tex]t^{2}[/tex] + 24t - 63
at turning point d/dt(h(t)) = 0
d/dt (h(t)) = -2t +24
-2t +24 = 0
if we differentiate again, second derivative is -2 which is negative, so it is a maximum point.
2t = 24
t = 12 seconds
h(t) at t = 12
h(t) = [tex]-(12)^{2}[/tex] + 24(12) - 63 = 81 feet
In conclusion, the maximum height of the airplane after 12 seconds is 81 feet.
Learn more about minimum and maximum points: brainly.com/question/14993153
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