Suppose a certain manufacturing company produces connecting rods for $4-$ and $6-$cylinder automobile engines using the same production line. The cost required produce the $4-$cylinder connecting rods is $$2,100,$ and the cost required to set up the production line for the $6-$cylinder connecting rods is $$3,300.$ Manufacturing connecting rod and $$16$ for each $6-$cylinder connecting rod. Hawkins makes a decision at the end of each week as to which product will be manufactured the follon is necessary from one week to the next, the weekend is used to reconfigure the production line. Once the line has been set up$,$ the weekly production capacities ar and $8,0004-$cylinder connecting rods. Let

$x_4=$ the number of $4-$cylinder connecting rods produced next week

$x_6=$ the number of $6-$cylinder connecting rods produced next week

$s_4=1$ if the production line is set up to produce the $4-$cylinder connecting rods; $0$ if otherwise

$s_6=1$ if the production line is set up to produce the $6-$cylinder connecting rods; $0$ if otherwise

$($a$)$ Using the decision variables $x_4$ and $s_4,$ write a constraint that limits next week's production of the $4-$cylinder connecting rods to either $0$ or $8,000$ units.

$($b$)$ Using the decision variables $x_6$ and $s_6,$ write a constraint that limits next week's production of the $6-$cylinder connecting rods to either $0$ or $5,000$ units.

$($c$)$ Write a third constraint that, taken with the constraints from parts $($a$)$ and $($b$),$ limits the production of connecting rods for next week.

$($d$)$ Write an objective function for minimizing the cost of production for next week.