Apple Tarts And Pies A Baker's Mathematical Constraints

Baking is an art, but beneath the surface of flaky crusts and sweet fillings lies a world of mathematical precision. Consider the case of a baker who crafts delectable apple tarts and pies each day. This seemingly simple scenario opens a door to explore systems of inequalities, a fundamental concept in mathematics. In this article, we'll delve into the baker's daily apple predicament, unraveling the constraints and formulating a system of inequalities that governs their baking endeavors. Understanding this system not only sheds light on the baker's challenges but also provides a practical application of mathematical principles.

The Baker's Apple Predicament

Our baker is a master of two apple-centric delights apple tarts and apple pies. Each tart, represented by the variable t, requires a single apple, while each pie, denoted by p, demands a generous eight apples. The baker's raw material, the apples, arrives in a daily shipment of 184. This is a crucial constraint the baker cannot use more apples than they receive. Furthermore, our baker has a limit on tart production; they can bake no more than 40 tarts each day. This limitation could stem from oven space, ingredient availability beyond apples, or simply a preference to maintain a certain balance between tarts and pies. The question now is: how can we mathematically represent these constraints and the baker's baking possibilities?

Formulating the Inequalities

To translate the baker's situation into mathematical language, we need to construct a system of inequalities. Inequalities, unlike equations, allow us to express a range of possibilities rather than a single fixed solution. This is perfect for our scenario, as the baker has flexibility in how many tarts and pies they bake, as long as they adhere to the apple and tart constraints.

1. The Apple Constraint: The total number of apples used for tarts and pies cannot exceed the daily shipment of 184. Since each tart uses 1 apple and each pie uses 8 apples, we can express this constraint as:

   1t + 8p ≤ 184
   

   This inequality states that the sum of apples used for tarts (1t) and pies (8p) must be less than or equal to 184.

2. The Tart Constraint: The baker can make no more than 40 tarts per day. This is a straightforward constraint:

   t ≤ 40
   

   This inequality limits the number of tarts (t) to a maximum of 40.

3. Non-Negativity Constraints: It's implicitly understood that the baker cannot make a negative number of tarts or pies. We express these constraints as:

   t ≥ 0
   p ≥ 0
   

   These inequalities ensure that both t and p are non-negative values.

The System of Inequalities

Combining these individual inequalities, we arrive at the complete system of inequalities that represents the baker's daily baking constraints:

1t + 8p ≤ 184
t ≤ 40
t ≥ 0
p ≥ 0

This system of inequalities is a mathematical representation of the baker's apple tart and pie production limitations. Any combination of t (tarts) and p (pies) that satisfies all these inequalities represents a feasible baking plan for the baker.

Visualizing the Feasible Region

Systems of inequalities are not just abstract mathematical expressions; they have a visual representation that can provide valuable insights. By graphing the inequalities on a coordinate plane, we can identify the feasible region, which represents all possible combinations of tarts and pies that the baker can produce while adhering to the constraints.

Graphing the Inequalities

To graph the inequalities, we first treat them as equations and plot the corresponding lines. Remember that inequalities represent a region of the plane, not just a line. The lines act as boundaries for these regions.

1. 1t + 8p ≤ 184: To graph this, we first graph the line 1t + 8p = 184. We can find two points on this line by setting t = 0 and solving for p (giving us the point (0, 23)), and setting p = 0 and solving for t (giving us the point (184, 0)). Connect these points to draw the line. Since the inequality is less than or equal to, we shade the region below the line, representing all points that satisfy the inequality.

2. t ≤ 40: To graph this, we first graph the line t = 40, which is a vertical line passing through the point (40, 0). Since the inequality is less than or equal to, we shade the region to the left of the line.

3. t ≥ 0 and p ≥ 0: These inequalities restrict our solutions to the first quadrant of the coordinate plane, as both t and p must be non-negative.

Identifying the Feasible Region

The feasible region is the area where all the shaded regions from the individual inequalities overlap. This region is a polygon bounded by the lines we graphed. Any point within this polygon or on its boundaries represents a combination of tarts (t) and pies (p) that the baker can make without exceeding their apple supply or tart production limit.

Interpreting the Feasible Region

The feasible region provides a visual representation of the baker's options. For instance, a point near the top of the region represents a scenario where the baker makes many pies but few tarts. A point near the right edge represents making the maximum number of tarts, while a point in the middle represents a balance between tart and pie production. Understanding the feasible region allows the baker to quickly assess the possibilities and make informed decisions about their daily baking plan.

Optimizing the Baker's Output

Now that we've established the constraints and visualized the feasible region, we can delve into the question of optimization. Bakers, like any business owners, often strive to maximize their output or profit. Let's assume our baker makes a profit of $2 per tart and $5 per pie. How can they determine the optimal number of tarts and pies to bake to maximize their profit, given the constraints?

The Objective Function

To address this optimization problem, we need to define an objective function. The objective function represents the quantity we want to maximize or minimize. In this case, the baker wants to maximize profit. Let P represent the total profit. Since the profit per tart is $2 and the profit per pie is $5, our objective function is:

P = 2t + 5p

This equation expresses the total profit P as a function of the number of tarts t and pies p baked.

Finding the Optimal Solution

The optimal solution, the combination of tarts and pies that maximizes profit, lies within the feasible region. A crucial theorem in linear programming states that the optimal solution will always occur at a corner point (also called a vertex) of the feasible region. This significantly simplifies our search, as we only need to evaluate the objective function at these corner points.

1. Identify the Corner Points: The corner points are the points where the boundary lines of the feasible region intersect. We can find these points by solving the systems of equations formed by the intersecting lines. For our baker's problem, the corner points are (0, 0), (40, 0), (40, 18), and (0, 23).

2. Evaluate the Objective Function at Each Corner Point: We substitute the coordinates of each corner point into the objective function P = 2t + 5p and calculate the profit:

   • (0, 0): P = 2(0) + 5(0) = $0
   • (40, 0): P = 2(40) + 5(0) = $80
   • (40, 18): P = 2(40) + 5(18) = $170
   • (0, 23): P = 2(0) + 5(23) = $115

3. Determine the Maximum Profit: By comparing the profits calculated at each corner point, we can identify the maximum profit. In this case, the maximum profit is $170, which occurs when the baker bakes 40 tarts and 18 pies.

The Optimal Baking Plan

Therefore, to maximize their profit, the baker should bake 40 tarts and 18 pies each day. This solution satisfies all the constraints the apple supply, the tart production limit, and the non-negativity requirements and yields the highest possible profit. This optimization process demonstrates the power of mathematical tools in real-world decision-making.

Real-World Applications and Extensions

The apple tart and pie problem is a simplified example of a broader class of problems known as linear programming problems. Linear programming is a powerful mathematical technique used to optimize a linear objective function subject to linear constraints. It has a wide range of applications in various fields, including:

• Business and Economics: Optimizing production schedules, resource allocation, inventory management, and investment portfolios.
• Logistics and Transportation: Determining the most efficient routes for delivery trucks, optimizing warehouse locations, and scheduling airline flights.
• Manufacturing: Minimizing production costs, maximizing output, and optimizing the use of raw materials.
• Healthcare: Optimizing hospital staffing levels, scheduling patient appointments, and allocating medical resources.

Extensions to the Baker's Problem

Our baker's problem can be extended in several ways to make it more realistic and complex. For example:

• Multiple Ingredients: We could introduce additional ingredients, such as sugar and flour, with their own constraints and requirements.
• Varying Prices: The selling prices of tarts and pies could fluctuate depending on market demand.
• Time Constraints: The baker might have a limited amount of time to bake, with different baking times for tarts and pies.
• Demand Constraints: There might be a maximum number of tarts or pies that the baker can sell each day.

These extensions would add more complexity to the system of inequalities and the optimization process, but the fundamental principles of linear programming would still apply. By formulating the problem mathematically and using techniques like graphing and evaluating corner points, the baker could continue to make informed decisions and optimize their baking operations.

Conclusion

The baker's apple tart and pie dilemma beautifully illustrates the practical application of systems of inequalities and linear programming. By translating real-world constraints into mathematical expressions, we can gain valuable insights and make optimal decisions. From determining feasible baking plans to maximizing profit, the mathematical approach provides a powerful framework for problem-solving. This example serves as a reminder that mathematics is not just an abstract subject but a powerful tool that can be applied to a wide range of everyday situations, from the simple act of baking to complex business operations.