A platform support system is shown in the figure below. Platform one is subjected to a

distributed load $q_2($lb$/$ft$)$ and platforms two and three are subjected to a distributed load $q_1$

$($lb$/$ft$).$ Cable $1$ can support $200lb$ each, and cable $2$ can support $200lb.$ cable $3$ can support

$150lb,$ and cable $4$ can support $140lb$ each. Determine $q_1$ and $q_2$ such that the total load

supported by the system is maximized.

$(1)$ Using the given plot template $($next page$),$ find the optimum solution graphically.

$q_1^{**}=$

$,q_2^{**}=$

$,$ max load $=$

$(2)$ Without actually completing the Simplex method solution, answer the following

questions: If the Simplex method is used,

$($a$)$ In cycle one, the incoming basis variable is

and the outgoing variable

is

$($b$)$ Determine the values of the non$-$zero slack

variables