A company makes two products: $A$ and $B$. Each product $A$ gives \$30 in profit and uses 2 hours of labor; each product $B$ gives \$50 in profit and uses 3 hours of labor. The weekly labor available is at most 60 hours per week. They must make at least 5 units of product $A$. There is no minimum for $B$. Set up a system of inequalities and determine which corner of the feasible region (integer or fractional that is allowed) that will maximizes the profit: $P=30A+50B$. Show all calculations.

$(A,B)=(5,16)$ with a profit of $\$30\cdot 5+\$50\cdot 16=\$950$, because integer units are required in the production planning.

$(A,B)=(30,0)$ with a maximum profit of $\$900$, because using all the working hours on $A$ will always dominate the fractional solutions.

$(A,B)=(5,\frac{50}{3})$ with maximum profit $\frac{\$2950}{3}\approx \$983.33$.

No maximum exists because both variables are unbounded above, despite the labor constraint.