A rocket is shot into the air and follows a parabolic path given by h = -4.9t^2 + 58.8t + 8.6 where h is the height of the rocket above the ground in meters and t is the elapsed time in seconds. Determine how long the rocket will be above 150 m.
What is t when h = 150 ?
150 = -4.9 t^2 + 58.8 t + 8.6
or
4.9 t^2 - 58.8 t + 141.4 = 0
solve quadratic
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8.67 - 3.33 = time above 150
To determine how long the rocket will be above 150 meters, we need to find the value of t when the height of the rocket, h, is equal to 150 meters.
We have the equation for the height of the rocket: h = -4.9t^2 + 58.8t + 8.6
To find the time when the rocket is at 150 meters, we can substitute 150 for h in the equation and solve for t.
150 = -4.9t^2 + 58.8t + 8.6
To solve this equation, we can rearrange it to form a quadratic equation:
4.9t^2 - 58.8t - 141.4 = 0
Now we can solve this quadratic equation using the quadratic formula:
t = (-b ± √(b^2 - 4ac)) / (2a)
Here, a = 4.9, b = -58.8, and c = -141.4.
Plugging these values into the formula, we get:
t = (-(-58.8) ± √((-58.8)^2 - 4 * 4.9 * (-141.4))) / (2 * 4.9)
Simplifying further:
t = (58.8 ± √(3451.44 + 2772.08)) / 9.8
t = (58.8 ± √(6223.52)) / 9.8
t = (58.8 ± 78.85) / 9.8
We have two possible solutions:
t = (58.8 + 78.85) / 9.8 ≈ 14.29 seconds
t = (58.8 - 78.85) / 9.8 ≈ -2.04 seconds
Since time cannot be negative in this context, we discard the negative solution.
Therefore, the rocket will be above 150 meters for approximately 14.29 seconds.