With an initial velocity of 100 feet per second the function h(t) = -16t squared + 100t gives the height in feet of the ball after t seconds

Question

With an initial velocity of 100 feet per second the function h(t) = -16t squared + 100t gives the height in feet of the ball after t seconds

How long is the football in the air before it hits the ground

what is the maximum height reached by the football round to the nearest foot.

Answer 1

h=0 = -16t^2 + 100t

-4t(4t -25) = 0

t=0 and t=25/4 = 6 1/4 seconds

for 6 1/4 seconds, it's in the air before it hits the ground

or

take the derivative and set equal to zero, then double the time

-32t +100 = 0

t =100/32 = 25/8 = 3 1/8 to max height

twice that time to hit the ground

= 6 1/4 seconds

max height = h(3 1/8) = -16(3.125)^2 +100(3.125)

= -16(25/8)^2 +100(25/8)

= -16(625/64) + 100(25/8)

= -625/4 + 2500/8

=-1250/8 +2500/8

= 1250/8

= 625/4

= 156 1/4 feet = max height

156 ft to the nearest foot

or rewrite the height function in vertex form

the vertex will be the maximum point

where the x coordinate is the time at max height and

the y coordinate will= max height

h = -16t^2 +100t

= -16(t^2 -25t/4 + (25/8)^2)) +16(25/8)^2

= -16(t- 25/8)^2 + 625/4

vertex = (25/8, 625/4)

= (3 1/8, 156 1/4)

time at max height= half the time to reach the ground

time to reach the ground = 2 time 3 1/8 = 6 1/4

max height = 156 1/4 feet