This is a parabola. Zeroes (that is s(t) is zero ) when t=0, and t=4.
So the max height due to symettry of the parabola occurs at t=2.
So the max height due to symettry of the parabola occurs at t=2.
t = -b / (2a)
In this case, a = -16 and b = 64. Let's substitute these values into the formula:
t = -(64) / (2(-16))
t = -64 / (-32)
t = 2
So, the rocket will reach its maximum height at t = 2 seconds.
To find the maximum height, we substitute the value of t back into the equation s(t):
s(t) = 64t - 16t^2
s(2) = 64(2) - 16(2^2)
s(2) = 128 - 16(4)
s(2) = 128 - 64
s(2) = 64
Therefore, the rocket will reach a maximum height of 64 feet.
To find the time it takes for the rocket to hit the ground, we need to find the value of t when s(t) = 0. The equation for s(t) is already given as s(t) = 64t - 16t^2. Let's set it equal to zero and solve for t:
64t - 16t^2 = 0
t(64 - 16t) = 0
From this equation, we can determine that either t = 0 or 64 - 16t = 0.
For t = 0, the rocket hasn't been fired yet, so it is not the solution we are looking for.
To find the value of t when 64 - 16t = 0, we can solve for t:
64 - 16t = 0
16t = 64
t = 64 / 16
t = 4
Therefore, the rocket will hit the ground after 4 seconds.
In summary:
- The maximum height the rocket will reach is 64 feet.
- The rocket will hit the ground after 4 seconds.
In this case, the equation s(t) = 64t - 16t^2 represents the height of the rocket. Therefore, we can rewrite it as s(t) = -16t^2 + 64t.
Comparing this with the vertex form, we can see that a = -16, which means the parabola opens downwards. The vertex is given by the formula:
h = -b/2a
where b is the coefficient of the linear term (-16t).
So, substituting the values, we have:
h = -64 / 2*(-16)
h = -64 / -32
h = 2
Therefore, the rocket will reach its maximum height of 2 feet.
To find the time it takes for the rocket to hit the ground, we need to determine when the height reaches zero. We can set s(t) = 0 and solve for t.
-16t^2 + 64t = 0
Factoring out common terms, we get:
-16t(t - 4) = 0
Using the zero product property, we can set each factor equal to zero:
-16t = 0 or t - 4 = 0
Solving for t, we have:
t = 0 or t = 4
Therefore, the rocket will hit the ground in 4 seconds.