$.$ A manufacturing company produces two popular grades of commercial carpeting among its many other products. In the coming production period, management needs to decide how many rolls of each grade carpet should be produced in order to maximize profit.

$$ Each roll of Grade $1$ carpet uses $50$ units of synthetic fiber, requires $25$ hours of production time, and needs $20$ units of foam backing.

$$ Each roll of Grade $2$ carpet uses $40$ units of synthetic fiber, requires $28$ hours of production time, and needs $15$ units of foam backing.

$$ The profit per roll of Grade $1$ carpet is $$200$ and the profit per roll of Grade $2$ carpet is $$160.$

In the coming production period,

$$ the company has $3,000$ units of synthetic fiber available for use.

$$ Workers have been scheduled to provide at least $1,800$ hours of production time but no more than $2,000$ hours of production time.

$$ The company has $1,500$ units of foam backing available for use.

a$.$ What are your decision variables $($x$)?$ State the units for each.

b$.$ Write down your objective function and constraints below, include brief descriptions$/$explanations on the right.

Delete or add rows as needed. Don$$t forget the nonnegativity constraint.

s$.$t$.($subject to$$ the following constraints$)$

c$.$ What if marketing required that the ratio of Grade $1$ carpet to Grade $2$ carpet had to be at least $3$ to $5.$ Write down this LP constraint below $($make sure it is in linear form & all variables are on the left$-$hand$-$side of the constraint$).$