Resource allocation $($Throwback to linear programming$)$

A company produces Four types of product lines, A$,$ B and C and D$.$ It has $3$ different types of machines on which

to create these products. The number of machines, profit per unit and amount of hours it takes to create the

product on the machine are listed below. The machines may be used for up to fifty hours per week to create the

products.

Assuming that every product made will be sold and that fractional products can be sold:

a$)$ Define the control variables for a linear programming model.

b$)$ Maximize the profit that the company can realize using a linear programming model.

c$)$ Identify the number of products that will maximize the profit.

d$)$ If the company decided to change the price to increase their profit margin on Product $C$ to $$14$

would that change the level of production that optimizes profit? Justify your answer through

sensitivity or a new model.