How do you solve quadratic word problems?

Question

A football is thrown by a quarterback and the height of the ball, h(t), at time t is determined by the equation h(t) = -16t^2 + 64t + 4. For which interval of time is the height of the football greater than or equal to 4 feet. This is not hw but I have a test tomorrow and need to know how to do it.

Answer 1

h(t) = -16t^2 + 64t + 4
4 - 4 <= -16t^2 + 64t + 4 - 4
0/t <= (-16t^2 + 64t)/t
0 <= -16t + 64

The beginning of the interval is therefore t = 0, since we were able to divide by t there.

0 <= -16t + 64
0 - 64 <= -16t + 64 - 64
-64 <= -16t
-64/-16 <= -16t /-16
t = 4 sec

Thus, the football is over 4 feet from 0 to 4 seconds.

Now, there's a more likely scenario, where you have to use the quadratic formula. For example, to find out what interval of time the height was greater than 2 feet:
h(t) = -16t^2 + 64t + 4
2 - 2 <= -16t^2 + 64t + 4 - 2
0 <= -16t^2 + 64t + 2
t = (-b +/- (b^2 - 4ac)^1/2)/2a
t = (-64 +/- (4096 + 128)^1/2)/-32
t = 2 +/- 64.99/-32
t= 2 +/- 2 1/32 = -1/32, 4 1/32
Ignore the negative solution, so the ball was at or higher than 2 feet from 0 to 4 1/32 seconds.

Answer 2

two ways.

mathematically:

change formula to -16x^2 + 64t + 4 is greater or equal to 4

graphically (calculoator or hand):

graph -16x^2 + 64t + 4 and graph 4 (should be a horizontal line). the exponential graph will eventually rise above the line 4 and then back below it. the interval for which it is above 4 feet is the x of the first intersection to the x of the second.