? You better give me a heart? a) To find how long it will take for the rocket to return to the ground, you need to determine when its height (h) is equal to zero. You can do this by setting the equation for height equal to zero and solving for time (t):
h = -16t^2 + 122t
0 = -16t^2 + 122t
Now, you can solve this quadratic equation for t. You can either use the quadratic formula or factor it if possible. Let's use the quadratic formula:
t = [-b ± √(b^2 - 4ac)] / (2a)
In this case, a = -16, b = 122, and c = 0.
t = [-122 ± √(122^2 - 4(-16)(0))] / (2(-16))
Now, calculate the values:
t = [-122 ± √(14984)] / (-32)
t = [-122 ± 122] / (-32)
Now, you have two potential solutions:
1. t = (−122 + 122) / (-32) = 0 seconds (The rocket was on the ground initially).
2. t = (−122 - 122) / (-32) ≈ 7.625 seconds (The rocket returns to the ground after approximately 7.625 seconds).
So, it will take approximately 7.625 seconds for the rocket to return to the ground.
b) To find when the rocket is 110 feet above the ground, you can set h equal to 110 and solve for t:
h = -16t^2 + 122t
110 = -16t^2 + 122t
Now, solve this quadratic equation for t. Again, you can use the quadratic formula:
t = [-b ± √(b^2 - 4ac)] / (2a)
In this case, a = -16, b = 122, and c = -110.
t = [-122 ± √(122^2 - 4(-16)(-110))] / (2(-16))
Calculate the values:
t = [-122 ± √(14984 + 7040)] / (-32)
t = [-122 ± √(22024)] / (-32)
t = [-122 ± 148.297] / (-32)
Now, you have two potential solutions:
1. t = (-122 + 148.297) / (-32) ≈ 0.893 seconds (rounded to 3 decimals).
2. t = (-122 - 148.297) / (-32) ≈ 8.881 seconds (rounded to 3 decimals).
So, the rocket will be approximately 110 feet above the ground at t ≈ 0.893 seconds and t ≈ 8.881 seconds.