Thank you for sharing the image! It contains several mathematical problems, mostly related to optimization and linear programming.

Here's a breakdown of the content:

Problem 8 (Linear Programming)

The problem asks to model the scenario of a cake maker who will make two types of cakes, A and B. It involves the constraints on the amount of butter and flour available, alongside a mathematical model. The options provided for the mathematical model are:

A. $10x + 7y \leq 500; \, 2x + 3y \leq 500; \, x \geq 0; \, y \geq 0$ B. $7x + 10y \leq 800; \, 3x + 2y \leq 500; \, x \geq 0; \, y \geq 0$ C. $7x + 10y \leq 500; \, 3x + 2y \leq 500; \, x \geq 0; \, y \geq 0$ D. $7x + 10y \leq 500; \, 5x + 3y \leq 500; \, x \geq 0; \, y \geq 0$ E. $7x + 10y \leq 500; \, 5x + 3y \leq 500; \, x \geq 0; \, y \geq 0$

Problem 9 (Optimization)

You are tasked with finding the maximum value of the objective function $f(x, y) = 3x + 4y$ based on the given system of inequalities.

The given system is:

• $12x + 3y \leq 36$
• $2x + y \geq 10$
• $x \geq 0$
• $y \geq 0$

You are asked to find the maximum value of the function $f(x, y)$ that satisfies the system.

Problem 10 (Graph Analysis)

This problem involves analyzing a graph with a region shaded in the x-y plane. Based on the graph, you need to find the maximum value of the objective function $f(x, y) = 3x + 2y$ over the feasible region defined by the constraints. The question provides the following options for the maximum value:

A. 6
B. 10
C. 11
D. 15
E. 16

Would you like help solving one or more of these problems in detail? Let me know which question you'd like to start with!