A ball is thrown into the air. The height in feet of the ball can be modeled by the equation h = -16t2 + 20t + 6 where t is the time in seconds, the ball is in the air. When will the ball hit the ground? How high will the ball go?
16 t^2 -20 t - 6 = - h in obsolete English units
when does h = 0?
t = [ 20 +/- sqrt (400 + 384) ] /32
t = [ 20 +/- 28 ] /32
t = 48/32 = 1.5 seconds (ignore the negative t)
for vertex complete the square assuming no calculus may be used
16 t^2 -20 t = - h + 6
t^2 - (5/4) t = -h/16 + 3/16
t^2 - (5/4)t + 25/64 = -4h/64 + 37/64
(t-5/8)^2 = -(4/64) (h - 148 )
max height = 148 at t = 5/8 sec
Find the vertex of the parabola by completing the square:
h(t)=-16t²+20t+6
=-16(t²-2(5/8)t+25/64)+6+25/4
which means that the vertex is at
t=5/8
and it will hit the ground at h(t)=0, or
t=3/2 s (rejecting negative root).
Maximum is at the vertex, or
h(5/8)=49/4 ft.
thanx both of yall XD
To find out when the ball will hit the ground, we need to determine the value of "t" when the height "h" is zero. We can set the equation h = 0 and solve for t.
The given equation is: h = -16t^2 + 20t + 6
Setting h = 0, we have:
0 = -16t^2 + 20t + 6
This is a quadratic equation. To solve it, we can use the quadratic formula:
t = (-b ± sqrt(b^2 - 4ac)) / (2a)
In this case, a = -16, b = 20, and c = 6. Plugging in these values, we have:
t = (-20 ± sqrt(20^2 - 4(-16)(6))) / (2(-16))
Simplifying further:
t = (-20 ± sqrt(400 + 384)) / (-32)
t = (-20 ± sqrt(784)) / (-32)
t = (-20 ± 28) / (-32)
Splitting the equation into two parts:
t = (-20 + 28) / -32 ---> t = 8 / -32 ---> t = -1/4
t = (-20 - 28) / -32 ---> t = -48 / -32 ---> t = 3/2
Since time cannot be negative in this context, we discard the answer t = -1/4. Therefore, the ball will hit the ground at t = 3/2 or 1.5 seconds.
To determine the peak height of the ball, we can substitute this t value back into the equation h = -16t^2 + 20t + 6:
h = -16(3/2)^2 + 20(3/2) + 6
h = -16(9/4) + 30/2 + 6
h = -36/4 + 30/2 + 6
h = -9 + 15 + 6
h = 12
Therefore, the ball will reach a peak height of 12 feet.