A weight is attached to a vertical spring hung upside down is released and moves up and down periodically according to the function $h(t)=$Acos$(2ft)+c$ where $h$ is the height above the ground in metres, $A$ is the amplitude, $f$ is the frequency $($number of cycles per second$),t$ is the time, in seconds and $c$ is the axis $($half$-$way between the minimum and maximum heights$).$ Its maximum height above the ground is $1\mathrm{.}6$ m and its minimum height is $0\mathrm{.}8$ m above the ground. The weight completes $1$ cycle in $2$ seconds.

a$)$ Determine the equation for the height of the weight as a function of time.

$[2]$

b$)$ Find the equation that represents the vertical velocity of the weight as a function of time

$[1]$

c$)$ Find the equation that represents the vertical acceleration of the weight as a function of time.

d$)$ What is the maximum vertical velocity of the weight and at what time does this occur?

$[2]$

Note: Max velocity occurs when acceleration is $0.$

$[6]$