calculus
posted by fareha .
5. A particle moves along the xaxis in such a way that its position at time t is given by x=3t^416t^3+24t^2 for 5 ≤ t ≤ 5.
a. Determine the velocity and acceleration of the particle at time t.
b. At what values of t is the particle at rest?
c. At what values of t does the particle change direction?
d. What is the velocity when the acceleration is first zero?

a) velocity = x' = 12t^3  48t^2 + 48t
acc. = x'' = 36t^2  96t + 48
b) to be at rest, x' = 0
12t^3  48t^2 + 48t = 0
12t(t^2  4t + 4) = 0
12t(t2)^2 = 0
t = 0 or t = 2
c) mmmh, what must have happened to the velocity when direction is changed?
d) for acc. = 0, 36t^2  96t + 48 = 0
3t^2  8t + 4 = 0
(t2)(3t  2) = 0
t = 2 or t = 2/3
so when t = 2/3, sub that into x'
I will let you do the arithmetic.