A football is kicked. It's height above ground is modelled by h=30t-6t^2, h= height in meters, t= time in seconds

Question

What time does the football hit the ground? What is the maximum height it reaches and what time does it reach it? 

Answer 1

h(t) = 30t - 6t²

 

This is a parabola opening downward.  We want to know when the ball will hit the ground

so we are solving to find when h(t) = 0 so we have:

 

-6t² + 30t = 0

We can factor the -6t out of this so we have

-6t(t-5) = 0

 

This is true for t=0 and t=5

The ball starts at ground level before being kicked, then hits the ground again at t = 5 seconds

 

For the maximum height we need to know the y value of the vertex for the parabola

 

At the vertex of the parabola the x value is found by x=-b/2a from the quadratic equation.

This equation is in the form h(t) = at² + bt + c = 0

so a = -6, b = 30, c=0

 

To find -b/2a  just plug in    -b/2a = -30/(-12)  = 5/2

So the x coordinate (time) of the vertex is 5/2

 

Plug that value into the original equation to find the y coordinate (height) of the vertex

 

-6t² + 30t  with t = 5/2

-6(5/2)²+30(5/2) = -6(25/4) + 75 = 75/2 = 37.5m

 

The vertex is located at (2.5, 37.5)  and the maximum height reached is 37.5m