What is the maximum height that the flare reaches

Question

The height above the ground of a launched object is a quadratic function of the time that it is in the air. Suppose that a flare is launched from a cliff 64 ft above sea level. If 3 sec after being launched the flare is again level with the cliff and if 12 sec after that it lands in the sea, what is the max height that the flare reaches? 

Answer 1

h(t) = At² + Bt + 64, t in seconds, h in feet above sea level

 

h(3) = 64 ⇒ 64 = A(3²) + B(3) + 64

9A + 3B = 0

3A + B = 0

B = -3A

 

I assume "12 sec after that" is 12 seconds after it was at 64 feet, which is 3 + 12 = 15 seconds after launch.  (Or did you mean 12 seconds after launch?)

 

h(15) = 0 = A(15²) - 3A(15) + 64

A = -64 / (225 - 45) ≅ -0.3556

 

B = -3A = -3(-0.3556) = 1.0668

 

h(t) = -0.3556t² + 1.0668t + 64

 

Maximum height is at the vertex of the parabola, which is at time

 

t = -B/(2A) = -1.0668 / [2(-0.3556)] = 1.5 sec

 

h_max = h(1.5) =  -0.3556(1.5)² + 1.0668(1.5) + 64

 

h_max ≅ 64.8 feet above sea level