a. To determine the altitude of the rock climber 2 hours after she begins her climb, we substitute t = 2 into the equation for a(t):
a(2) = -10(2)^2 + 60(2) = 80
Therefore, the altitude of the rock climber 2 hours after she begins her climb is 80 meters.
b. To determine the altitude of the rock climber 3 hours after she begins her climb, we substitute t = 3 into the equation for a(t):
a(3) = -10(3)^2 + 60(3) = 90
Therefore, the altitude of the rock climber 3 hours after she begins her climb is 90 meters.
c. To determine the average rate of change of the altitude of the rock climber between 2 and 3 hours after she begins her climb, we need to calculate the difference in altitude over the difference in time:
average rate of change = (a(3) - a(2)) / (3 - 2) = (90 - 80) / 1 = 10
Therefore, the average rate of change of the altitude of the rock climber between 2 and 3 hours after she begins her climb is 10 meters per hour.
d. To determine the instantaneous rate of change of the altitude of the rock climber 3 hours after she begins her climb, we need to take the derivative of the altitude function with respect to time:
a'(t) = -20t + 60
Substituting t = 3, we get:
a'(3) = -20(3) + 60 = 0
Therefore, the instantaneous rate of change of the altitude of the rock climber 3 hours after she begins her climb is 0 meters per hour.
e. The instantaneous rate of change value found in part d) represents the slope of the tangent line to the altitude function at t = 3, which corresponds to the rock climber's velocity at that instant. Since the instantaneous rate of change is 0, this tells us that the rock climber's velocity at that instant is 0, i.e., the rock climber has stopped moving.