Quadratic Word Problem?

Question

A football is kicked following the path modelled by the equation h= -4t^2 + 10t + 3, where h is the hight of the ball above the ground in metres and t is the time since the ball was kicked in seconds.
a) describe the path of the ball
b) after how many seconds does the ball reach the ground?
c) how long does the ball stay above 5m?

Please briefly describe a and b as I think I can answer these but what I need to know is why when graphing this equation is the x-coordinate less than zero? Secondly, I have no idea how to preform the operations to answer c....Thanks ahead of time!!

Answer 1

well a) is a parabolic arc and they probably want you to figure the max height, aka the max possible value of h for any t.
for b) the ball reaching the ground is h=0 so set the eq = 0 and solve for the factors that give values of t that make the eq true. one will be negative which is irrational (negative time) and one will be your answer.
for c) if you solve for h=5 then you can figure the time t that passes from h=0 to h=5. the ball is above 5m for a period of time that is equal to total time ( part b) minus the two periods that the ball is travelling up to 5 m and down from 5m.

Answer 2

a) The path of the ball is a parabolic arc since it's of the form:
h = at^2 + bt + c. In particular since a = - 4, (negative) then the arc is convex.
b) The ball reaches the ground means h = 0,
so solve for t in -4t^2 + 10t + 3 = 0. I think you have to use the quadratic formula. One answer will be negative. This is because when t = 0 h = 3, which means when kicked the ball was already above ground 3 meters, which is pretty strange.
The neg. time indicates how much time before 0 the ball would have had to been kicked at ground level and following the same arc.
c) So, put in 5 for h. You get -4t^2 + 10t + 3 = 5 or
-4t^2 + 10t - 2 = 0 or 2t^2 - 5t +1 = 0. Solve for t and this time you get 2 positive values. The smaller one indicates the time the ball is at 5 and going up, the second one, again at 5 but this time going down. Between those two times the ball is above 5, so take the difference of the two times.