Module calculus problem

Question

Given the position function,  , between t = 0 and t = 15, where s is given in feet and t is measured in seconds, find the interval in seconds where the particle is moving to the right.

Options:

1. 0<t<6
2. 5<t<15
3. t>15
4. The particle always moves to the right.

Answer 1

If "the particle is moving to the right" then the velocity will be positive (a zero velocity means the object isn't moving and a negative velocity means its moving left).

The velocity is the derivative of position with respect to time. So, taking the derivative, we get s'(t) = t² - 12t + 36 or, factoring, s'(t) = (t - 6)² which is positive for all values of t except it is zero for t = 6. So, in reality, none of the answers is correct but answer 4 is the closest (and is OK if you ignore the fact that it slows to a stop at t = 6 then speeds up again after that.