Exercise 16 Page 161

We will begin by organizing the given information in a table. Let $x$ and $y$ be the unknowns where $x$ is the number of corn muffins and $y$ be the number of bran muffins.

	Corn Muffins, $x$	Bran Muffins, $y$
Number of muffins	$x$	$y$
Number of cups of milk	$4x$	$2y$
Number of cups of wheat flour	$3x$	$3y$
Profit ($\$$)	$3x$	$2y$

Let's write the constraints!

Writing the Constraints

We can write the first two constraints based on the fact that the number of trees cannot be negative. Since the baker has $16$ cups of milk, the total number of cups of milk will be less than or equal to $16.$ The available wheat flour is $15$ cups. Therefore, the number of cups of wheat flour to make muffins will be less than or equal to $15.$

Graphing the Feasible Region

Let's start with the first two constraints. We will first determine their boundary lines by replacing the inequality sign with the equals sign. The boundary lines of the first two constraints are the axes. Since both $x$ and $y$ are greater than or equal to $0,$ we will shade Quadrant I. Notice that the inequalities are non-strict, so the boundary lines will be solid. Next, we will graph Constraint III and IV by finding their boundary lines. To graph their boundary lines, we will find the $x\text{-}$ and $y\text{-}$intercepts of the lines. We will first substitute $0$ for $x$ into Boundary Line III and solve it for $y$ to find its $y\text{-}$intercept. The $y\text{-}$intercept of Boundary Line III is the point $(0,8).$ Proceeding in the same way, we can find the intercepts of both lines.

Constraint	Boundary Line	$x=0$	$y\text{-}$intercept	$y=0$	$x\text{-}$intercept
III	$2x+y=8$	$2(0)+y=8$	$(0,8)$	$2x+(0)=8$	$(4,0)$
IV	$x+y=5$	$(0)+y=5$	$(0,5)$	$x+(0)=5$	$(5,0)$

Now we will plot the intercepts and connect them with line segments. The line segments will be bound by the axes because the number of muffins cannot be negative. Also, notice that the constraints are non-strict. Therefore, the line segments will be solid.

Next, we will test the point $(0,0)$ to determine which region we should shade. Let's start with Constraint III. For Constraint III, we will shade the region that contains the point. Now we will test the point for Constraint IV. Then, we will again shade the region that contains the test point.

The overlapping section of the graph above is the feasible region. Let's remove the unnecessary parts and indicate the vertices of the feasible region.

Objective Function and Maximum Profit

Let $P$ be the total profit. Using the fourth line of our table, we can write the objective function. To find the number of each type of muffin, we will look for the vertex that maximize the profit. Let's begin by substituting the vertex $(0,0)$ into the objective function. For the vertex $(0,0),$ the profit will be $\$0.$ We can find the profit for the other vertices proceeding in the same way.

Vertex	Objective Function	Profit ($\$$)
$(0,0)$	$P=3(0)+2(0)$	$0$
$(0,5)$	$P=3(0)+2(5)$	$10$
$(3,2)$	$P=3(3)+2(2)$	$13$
$(4,0)$	$P=3(4)+2(0)$	$12$

As a result, the baker should make $3$ corn muffins and $2$ bran muffins to maximize the profit.