A cart speed varies as v = 2500 t^2/3 mile/hour. How long does it take the cart to have a constant speed of 10 mile/hour if the cart start from rest?

Let the speed of the cart and the time be v and t respectively.

Since the speed is a positive value, the speed with respect to time is a one-to-one function as shown in the diagram on the left. Rearranging the function in terms of v gives

     t = v^3/2/2500

When the speed is 10 mile/hour, the time is

     t = 10^3/2/2500 = 0.0126 hour = 30.36 s