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**HallsofIvy** The rate of change of a function is its derivative. Saying that "The rate of change R, in km/hr, of the altitude of a hot air balloon is given by $\displaystyle R(t)= t^3= 4t^2+ 6$" means that, taking h(t) to be the altitude (in km) at time t (in hours), $\displaystyle \frac{dh}{dt}= t^3- 4t^2+ 6$. Take the "anti-derivative" to find h(t). Of course, that will involve a arbitrary "constant of integration". Use "Assume the balloon is initially at ground level" to find that constant.