$1.$ A mass is hanging at rest from a spring $30$ cm above a pressure plate as shown below. It is pulled down some distance and then released so that it starts Level oscillating $($bouncing$)$ up and down according to the equation:

d$($t$)=-8$cos$(180$t$)+30,$ where d$($t$)$ is the distance from the mass to the pressure plate at time t$,$ measured in seconds.

a$)$ What is the period of the function, and what does it represent in this situation? 

b$)$ What is the amplitude of the function, and what does it represent in this situation?

c$)$ What is the range of this function and what does it represent in this situation?

d$)$ Within the first $4$ seconds, at what times is the mass $32$ cm above the pressure plate?