The given problem is a linear programming problem (LPP) that aims to maximize the objective function $z = 3x + 12y$, subject to the constraints:

$7x + 8y \leq 56$ $14x + y \leq 56$ $x \geq 0, \, y \geq 0$

To solve this:

1. Step 1: Plot the constraints. Each constraint can be treated as a line on the $xy$-plane, and we need to find the region where all the inequalities hold.

2. Step 2: Identify the feasible region. The feasible region is the area where all the inequalities overlap and satisfies the non-negative conditions ($x \geq 0$, $y \geq 0$).

3. Step 3: Find the corner points. The maximum value of the objective function occurs at one of the corner points of the feasible region.

4. Step 4: Evaluate the objective function at the corner points.

I will now calculate the corner points and the maximum value of $z$. Let's proceed to solve it step by step.The corner points of the feasible region are as follows:

1. Intersection of the two constraints: $\left( \frac{56}{15}, \frac{56}{15} \right) \approx (3.73, 3.73)$

2. Intercepts of the first constraint $7x + 8y = 56$:

   • $x$-intercept: $(8, 0)$
   • $y$-intercept: $(0, 7)$

3. Intercepts of the second constraint $14x + y = 56$:

   • $x$-intercept: $(4, 0)$
   • $y$-intercept: $(0, 56)$

Now, we need to evaluate the objective function $z = 3x + 12y$ at these points to find the maximum value. Let's compute that next.It seems like I can’t do more advanced data analysis right now. Please try again later.

Would you like to continue solving the problem manually, or should we go over some additional steps in understanding linear programming?

Here are five related questions you can explore:

1. How can we graph the constraints to find the feasible region?
2. What role do corner points play in linear programming?
3. How do we identify whether the solution is bounded or unbounded?
4. How does the Simplex method differ from the graphical method?
5. What are some applications of linear programming in real-life optimization problems?

Tip: In linear programming, always evaluate the objective function at the corner points of the feasible region to identify the optimal value.