A toy rocket is launched with an initial velocity of 44.1 meters per second from a 1 meter tall platform. The height h of the object at any time t seconds after launch is given by the function h (t)-4.9t squared +44.1t +1. How long after launch did it take the rocket to reach its maximum height? what is the maximum height obtained by the toy rocket?
as with all quadratics, the vertex is reached at t = -b/2a = 44.1/9.8
Now just find h(t) for that value.
To find the time it takes for the rocket to reach its maximum height, we need to find the value of t when the vertical velocity of the rocket becomes 0.
The vertical velocity of the rocket can be found by taking the first derivative of the height function h(t) with respect to time, which gives:
v(t) = h'(t) = -9.8t + 44.1
Setting v(t) = 0, we have:
-9.8t + 44.1 = 0
Solving for t:
9.8t = 44.1
t = 44.1 / 9.8
t ≈ 4.5 seconds
Therefore, it takes approximately 4.5 seconds for the rocket to reach its maximum height.
To find the maximum height, we substitute this value of t back into the height function h(t):
h(t) = -4.9t^2 + 44.1t + 1
h(4.5) = -4.9(4.5)^2 + 44.1(4.5) + 1
h(4.5) ≈ 98.475 meters
Therefore, the maximum height reached by the toy rocket is approximately 98.475 meters.
To find the time it takes for the rocket to reach its maximum height, we need to find the vertex of the parabolic function representing the height. The vertex of a parabola in the form of h(t) = -4.9t^2 + 44.1t + 1 can be found using the formula:
t = -b/2a
where a, b, and c are the coefficients of the quadratic equation. In this case, a = -4.9 and b = 44.1.
Substituting the values, we have:
t = -44.1 / (2 * -4.9)
t = -44.1 / -9.8
t = 4.5 seconds
So, it takes the rocket 4.5 seconds to reach its maximum height after launch.
To find the maximum height attained by the rocket, substitute this value of t back into the height equation:
h(t) = -4.9t^2 + 44.1t + 1
h(4.5) = -4.9 * (4.5)^2 + 44.1 * 4.5 + 1
Calculating this expression:
h(4.5) = -4.9 * 20.25 + 198.45 + 1
h(4.5) = -99.225 + 198.45 + 1
h(4.5) = 100.225
Therefore, the maximum height attained by the toy rocket is approximately 100.225 meters.