a model rocket is launched off the top of a platform. the height of the rocket is given by the function h(t)=-4.9t^2+86t+2.1 where h(t) is height in meters and t is time in seconds. when will the rocket's height be 200m? what is the maximum height of the rocket?
To find when the rocket's height is 200m, we set h(t) = 200 and solve for t:
-4.9t^2 + 86t + 2.1 = 200
-4.9t^2 + 86t -197.9 = 0
Using the quadratic formula, t = (-b ± √(b^2 - 4ac)) / (2a), where a = -4.9, b = 86, and c = -197.9, we can find the values of t:
t = (-86 ± √(86^2 - 4*(-4.9)*(-197.9))) / (2*(-4.9))
t ≈ (-86 + √(7396 + 3883.08)) / (-9.8) ≈ (-86 + √11279.08) / (-9.8)
Taking the positive value since time cannot be negative:
t ≈ (-86 + 106.2) / (-9.8) ≈ 20.2 / (-9.8) ≈ -2.067
Therefore, the rocket's height is 200m at approximately 2.067 seconds.
To find the maximum height, we can observe that the rocket reaches its highest point when its vertical velocity becomes 0. This occurs when the rocket is at the peak of its parabolic trajectory.
To find the time when the rocket reaches its maximum height, we find the t-coordinate of the vertex of the parabola. The t-coordinate of the vertex is given by t = -b / (2a), where a = -4.9 and b = 86:
t = -86 / (2*(-4.9)) ≈ -86 / (-9.8) ≈ 8.776
So, the rocket reaches its maximum height at approximately 8.776 seconds.
To find the maximum height, we substitute this time into the equation h(t):
h(8.776) = -4.9*(8.776)^2 + 86*(8.776) + 2.1 ≈ 374.156
Therefore, the maximum height of the rocket is approximately 374.156 meters.