Frontline Agricultural Processing Systems uses several ingredients to make wheat crackers. After several years of operations and testing, their scientists found high protein and carbohydrates in two of their ingredients, barley and corn. While an ounce of barley costs $$0\mathrm{.}25,$ an ounce of corn costs $$0\mathrm{.}46.$ While an ounce of barley provides $9$ mg of protein and $1$ mg of carbohydrates, an ounce of corn provides $7$ mg of protein and $5$ mg of carbohydrates. Recently, demand for wheat crackers has increased. To lower the overall cost of producing wheat crackers, Frontline Agricultural Processing Systems will want to know how many ounces of barley and corn to include in each box of wheat crackers to meet the minimum requirements of $160$ milligrams of protein and $40$ milligrams of carbohydrates.

$($a$)$ Write the optimal solution in the format $($# ounces of Barley, # ounces of Corn$)$

$($b$)$ What is the lowest limit unit cost for corn without changing the optimal production schedule? Round to $2$ decimal places.

$($c$)$ Problem Formulation: Write the carbohydrate constraint.

$($d$)$ List the constraints$)$ that is$/$ are nonbinding?

$($e$)$ If the carbohydrate requirement is increased by $12$ milligrams, what would be the new optimal cost? Round to $2$ decimal places.

$($f$)$ If the protein requirement is reduced by $12$ milligrams, what would be the new optimal cost? Round to $2$ decimal places.

$($g$)$ If the cost for an ounce of corn decreases by $$0\mathrm{.}35,$ the optimal solution will stay the same.

$($h$)$ What is the range of optimality for the cost of barley? Write your answer in the format $($Lower limit$,$ Upper limit$).$ Round to $2$ decimal places.

$($i$)$ Determine the minimum cost for Frontline Agricultural Processing Systems to produce wheat crackers.