Calculus (Related Rates)
posted by Anonymous .
The position of a particle moving in a straight line is given by s(t) = (e^(t))(cos(5t)) for t>0, where t is in seconds. If the particle changes direction at time T seconds, then T must satisfy the equation:
cos(5T) = 0
5T = arctan(1/5)
5e^(t) sin(5t) = 0
tan(5T) = 1/5
cos(5T) = 5
I know that a change in direction will be marked by a change from positive to negative or vice versa, but I don't understand the equations the question gives me. Could someone please talk me through this process to find the right answer?

you are correct, as far as you go. When it changes from pos to neg, it will be zero.
Note that the particle changes direction, not position. So, its velocity changes sign. The velocity is given by the derivative.
s = e^t cos5t
s' = e^t (cos5t  5sin5t)
so, since e^t is always positive, we need
cos5t  5sin5t = 0
tan 5t = 1/5