Final answer:
The total distance travelled by the rock when it reaches the ground is approximately 39.1 meters.
Step-by-step explanation:
The displacement function of the rock can be modeled by the equation f(t) = -2t^2 + 15.7t, where t is the time in seconds. To find the total distance travelled by the rock when it reaches the ground, we need to find the time when the rock hits the ground. The time at which the rock hits the ground can be found by setting the displacement function equal to 0:
-2t^2 + 15.7t = 0
Solving this quadratic equation gives us two possible solutions: t = 0 and t = 7.85 seconds. Since the rock is thrown up in the air and comes back down, we can ignore the t = 0 solution. Therefore, the rock hits the ground after approximately 7.85 seconds.
To find the total distance travelled by the rock, we need to calculate the area under the curve of the displacement function from t = 0 to t = 7.85 seconds. We can use integration to find the area under the curve:
Total distance = ∫(0 to 7.85) |f(t)| dt
Using the given displacement function, we substitute -2t^2 + 15.7t for f(t):
Total distance = ∫(0 to 7.85) |-2t^2 + 15.7t| dt
Integrating the function |-2t^2 + 15.7t| gives us:
Total distance = [-2/3t^3 + 7.85t^2] from 0 to 7.85
Plugging in the values, we get:
Total distance = [-2/3(7.85)^3 + 7.85(7.85)^2] - [-2/3(0)^3 + 7.85(0)^2]
Simplifying, we get:
Total distance ≈ 39.1 meters