Question

(this question has been blowing my mind) The rate of change of the altitude of a hot-air balloon is given by r(t) =t^3 - 4t^2 + 6 for 0 <(equal to) t <(equal to) 8. write, BUT DO NOT EVALUATE, the expression that represents the change in altitude of the balloon during the time that the altitude is decreasing. (calculator required)

Answer 1

The altitude is decreasing whenever the rate r(t) is negative.
So you want to find the time interval over which r(t) is negative.
Say r(t) is negative from t1 to t2, where t1 > 0 and t2 < 8.
Then the change in altitude is the integral of r(t) from t1 to t2.