To solve this problem, we need to find the minimum and maximum values of the objective function $f(x, y) = 8x + 15y$ subject to the given constraints:

Constraints:

1. $x \geq 0$
2. $12x + 24y \geq 288$ or equivalently $x + 2y \geq 24$
3. $12x - y \leq 144$
4. $x + 6y \leq 78$

Steps:

1. Graph the Constraints: These inequalities define a feasible region in the $xy$-plane. The feasible region is where all inequalities intersect.
2. Identify Corner Points: Find the points of intersection of these lines, as the minimum and maximum of the objective function will occur at one of these vertices.
3. Evaluate the Objective Function at Each Vertex: Substitute the coordinates of each vertex into $f(x, y) = 8x + 15y$ to find the values.
4. Select the Minimum and Maximum Values.

Solving Each Constraint for Intersection Points

1. Convert each inequality to an equation for intersection purposes:

   • $x + 2y = 24$
   • $12x - y = 144$
   • $x + 6y = 78$
   • $x = 0$ (boundary due to $x \geq 0$)

2. Solve these equations in pairs to find intersections.

Let me calculate the intersection points and evaluate the objective function at those points.The intersection points derived from the constraints are:

1. $\left(\frac{312}{25}, \frac{144}{25}\right) \approx (12.48, 5.76)$
2. $(-3, 13.5)$ (This point is outside the feasible region as $x \geq 0$)
3. $(0, 12)$
4. $\left(\frac{942}{73}, \frac{792}{73}\right) \approx (12.90, 10.85)$
5. $(0, -144)$ (This point is outside the feasible region as $y \geq 0$)
6. $(0, 13)$

Feasible Points

The feasible points within the constraints are:

• $(12.48, 5.76)$
• $(0, 12)$
• $(12.90, 10.85)$
• $(0, 13)$

Now, I’ll substitute these points into the objective function $f(x, y) = 8x + 15y$ to find the minimum and maximum values.The minimum and maximum values of the objective function $f(x, y) = 8x + 15y$ and their corresponding points are:

• Minimum Value: $180$ at the point $(0, 12)$
• Maximum Value: $265.97$ at the point $(12.90, 10.85)$

These values are rounded to the nearest hundredth as required.

Would you like further details or have any questions?

Here are five related questions to deepen your understanding:

1. How would the solution change if one of the constraints were different?
2. Can you explain why we only consider corner points for the optimal solution?
3. What happens if the feasible region is unbounded?
4. How would we approach this if more constraints were added?
5. Can you interpret the geometric meaning of the objective function value?

Tip: In linear programming problems, always check that all constraints are met by each vertex before evaluating the objective function.