Answer:
Greatest height: 45 meters
Time for greatest height: 3 seconds
Height after 5 seconds: 25 meters above the cliff
Time for height of 40 meters: 7.123 seconds
Height after 7 seconds: -35 meters (35 meters below the cliff)
Step-by-step explanation:
to find the maximum height, we need to calculate the derivative of y in relation to t and then find when dy/dt = 0:
dy/dt = 30 - 10t = 0
10t = 30
t = 3 seconds
In this time, the height is:
y = 30*3 - 5*3^2 = 45 meters
After 5 seconds, the height is:
y = 30*5 - 5*5^2 = 25 meters
The time for the height of 40 meters is:
40 = 30t - 5t^2
t^2 - 6t - 8 = 0
Using Bhaskara's formula, we have:
Delta = 6^2 + 4*8 = 68
sqrt(Delta) = 8.246
t1 = (6 + 8.246) / 2 = 7.123 seconds
t2 = (6 - 8.246) / 2 = -1.123 seconds (negative value for time is not valid)
So the time when the rocket reaches 40 meters is 7.123 seconds
After 7 seconds, the height is:
y = 30*7 - 5*7^2 = -35 meters
The rocket will be 35 meters below the cliff.
