SOLUTION: A projectile is launched from ground level with an initial velocity of v0 feet per second. Neglecting air​ resistance, its height in feet t seconds after launch is given by s=-16

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Question 1203587: A projectile is launched from ground level with an initial velocity of v0 feet per second. Neglecting air​ resistance, its height in feet t seconds after launch is given by s=-16t^2+v0t . Find the​ time(s) that the projectile will​ (a) reach a height of 288 ft and​ (b) return to the ground when v0=144 feet per second.
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(b) The projectile returns to the ground after
  
enter your response here ​second(s).
Answer by math_tutor2020(3816) About Me  (Show Source):
You can put this solution on YOUR website!

Part (a)

s = -16t^2 + v0*t
s = -16t^2 + 144t
288 = -16t^2 + 144t
0 = -16t^2 + 144t - 288
-16t^2 + 144t - 288 = 0

We can then factor
-16t^2 + 144t - 288 = 0
-16(t^2 - 9t + 18) = 0
-16(t - 3)(t - 6) = 0
t-3 = 0 or t-6 = 0
t = 3 or t = 6


Or we can use the quadratic formula
t+=+%28-b%2B-sqrt%28b%5E2-4ac%29%29%2F%282a%29

t+=+%28-144%2B-sqrt%28%28144%29%5E2-4%28-16%29%28-288%29%29%29%2F%282%28-16%29%29 Plugged in a = -16, b = 144, c = -288

t+=+%28-144%2B-sqrt%2820736+-+18432%29%29%2F%28-32%29

t+=+%28-144%2B-sqrt%282304%29%29%2F%28-32%29

t+=+%28-144%2B-++48%29%2F%28-32%29

t+=+%28-144%2B48%29%2F%28-32%29 or t+=+%28-144-48%29%2F%28-32%29

t+=+%28-96%29%2F%28-32%29 or t+=+%28-192%29%2F%28-32%29

t+=+3 or t+=+6


Either way, the two answers for part (a) are t = 3 and t = 6

The projectile reaches 288 ft at t = 3 seconds when going upward.
Upon its downward trajectory, it gets back to a height 288 ft at t = 6 seconds.

The projectile is above 288 ft for the interval 3 < t < 6. Otherwise, the projectile is at 288 ft or lower.

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Part (b)

s = -16t^2 + 144t
-16t^2 + 144t = s
-16t^2 + 144t = 0
-16t(t - 9) = 0
-16t = 0 or t-9 = 0
t = 0/(-16) or t = 9
t = 0 or t = 9

The projectile starts on the ground, so it makes sense that t = 0 is one solution.

The other solution is t = 9 which is when the rocket returns to the ground again.
The flight time is 9 seconds.