The rotation of rod OA about O is defined by the relation θ= 0.5e^ -0.8t*sin3πt, where θ and t are expressed in radians and seconds, respectively. Collar B slides along the rod so that its distance from O is r= 1 + 2t- 6t^2 + 8t^3, where r and t are expressed in feet and seconds, respectively. When t= 0.5 s, determine (a) the velocity of the collar

(b) the acceleration of the collar

(c) the acceleration of the collar relative to the rod.

An external consultant recommends that a plant installs a bank of capacitors for power factor correction. this will reduce the peak electrical demand charges by an average of 93 kw every month. the plant current pays $13 per kw in peak demand charges. the capacitor bank will include 223 kw of fixed capacitors, and 183 of variable capacitors. the fixed capacitors cost $59 per kw, and the variable capacitors will cost $65 per kw. the consultant charges 21% of the equipment costs to install the capacitors. because this project will reduce the demand for the electric utility, they are prepared to provide a one-time rebate of $42 per kw of reduced demand. what is the simple payback period for this project (in years)?