A bottle rocket is launched at an initial velocity of 85 feet per second from a platform that is 2 feet above the ground. Its height (h) in feet at a given the seconds since launch (t) can be modelled with the function
h(t)=-16t^2+85+2
Determine how long it will take for the rocket to hit the ground. Round your answer to one tenth of a second, if needed.
t=--- seconds
To determine how long it will take for the rocket to hit the ground, we need to find the value of t when h(t) = 0.
Setting h(t) = 0, we have:
-16t^2 + 85 + 2 = 0
Simplifying this equation, we get:
-16t^2 + 87 = 0
To solve for t, we can divide both sides of the equation by -1:
16t^2 - 87 = 0
Now, we can solve this quadratic equation by factoring or by using the quadratic formula. Factoring doesn't seem to be feasible in this case, so let's use the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / 2a
For our equation, a = 16, b = 0, and c = -87. Substituting these values into the quadratic formula, we have:
t = (0 ± √(0^2 - 4(16)(-87))) / 2(16)
= (0 ± √(0 + 5568)) / 32
= (± √5568) / 32
Taking the positive square root, we have:
t = √5568 / 32
t ≈ 5.92 seconds
Therefore, it will take approximately 5.92 seconds for the rocket to hit the ground.