quadratic functions help please

Question

the formula h=-16t^2 +vt if the model rocket is fired at an intial vertical velocity of 78 meteres per second will the rocket reach a height of 80 meters? t

Answer 1

The formula h=-16t^2 +v0t if the model rocket is fired at an initial vertical velocity of 78 meters per second will the rocket reach a height of 80 meters?

h = -16t^2 + 78t = -2t(8t - 39)

Zeros: t = 0 and t = 39/8

Axis of Symmetry and Vertex at t = 39/8/2 = 39/16.

h(39/16) = -2(39/16)(8(39/16) - 39)

= -(39/8)(39/2 - 39)

= -(39/8)(-39/2) = 39^2/4^2 = (39/4)^2

= ((40-1)/4)^2 = (10 - 1/4)^2

= 100 - 5 + 1/16 = 95 + 1/16 > 80

Yes, the rocket will reach and exceed a height of 80 meters.

Answer 2

h = -16t² + 48t   (v = initial velocity of 48 ft/sec)

 

Since the coefficient of the t² term is negative (-16), the parabola opens downward and its vertex will be the highest point.  The t-coordinate of the vertex is:

 

t(vertex) = -b/2a

 

Where a is the coefficient of the t² term (-16) and b is the coefficient of the t term (48).  Plug in the values to find the time it takes the rocket to reach its highest point, then plug that time into the h equation to determine the highest height.