The table gives dimensions for two different four bar linkages. Link numbering is defined as follows:

Link $1-$ Ground link

Link $2-$ Input link

Link $3-$ Coupler link

Link $4-$ Output link

For each linkage, explore the range of motion of the transmission angle and input angle. All angles are defined in the figure. Specifically, determine the following:

$($i$)$ Determine the range of the transmission angle $.$ If it will not reach $180$ or $0,$ then calculate the maximum and$/$or minimum transmission angle. Express your answers in the positive range Sketch the linkage in the position$($s$)$ it occupies when these extreme$($s$)$ occur.

$($ii$)$ Determine range of the input angle ${}_2={}_{12}.$ If it will not reach $180$ or $0,$ then calculate the maximum and$/$or minimum input angle. Express your answers in the positive range Sketch the linkage in the position$($s$)$ it occupies when these extreme$($s$)$ occur. For clarity, sketch a pie chart approximately to scale that shows the entire possible range of the input angle, including all circuits.

$\backslash$table$[[$Problem$,$Link $1,$Link $2,$Link $3,$Link $4],[$a$,8,4,7,6],[$b$,6,4,8,11]]$