The displacement of an oscillating mass is given by x(t) = 20 cos( 5 t ). at time t = 0.5π s, what is the acceleration of the mass in m/s2?

The speed of the object is the 1st derivative of its displacement.

The acceleration is the 2nd derivative of the displacement.  That's also the first derivative of speed.  Either way you approach it, we have to take the displacement and differentiate twice.

Displacement = x(t) = 20 cos(5t) meters

Speed = x'(t) = -100 sin(5t) m/s

Acceleration = x''(t) = -500 cos(5t)  m/s²

At time  t=0.5π sec,

Acceleration = -500 cos(2.5π)  m/s²

cos (2.5π) = cos(0.5π) = zero

This is weird.  At  t=0.5π sec, the acceleration of the oscillating mass appears to be zero.

My math must be kerflooie, but I'm too tired right now to stay and pick at it.  I'll just resign myself to having my answer reported, and I hope whoever reports it will be kind enough to point out the error of my ways.

c. magnitude and direction

see the picture for the vector

the magnitude of the vector by pythagorean theorem

r = √( (-10)^2 + (3)^2 ) = √109

magnitude = √109 units

the direction angle between vector and x-axis is

tan^(-1)(3/10) = 16.699

so the direction is approximately 16.7 degrees north of west