A Baker Makes Apple Tarts And Apple Pies Each Day. Each Tart, { T $}$, Requires 1 Apple, And Each Pie, { P $}$, Requires 8 Apples. The Baker Receives A Shipment Of 184 Apples Every Day. If The Baker Makes No More Than 40 Tarts Per

Mar 8, 2025 by ADMIN 231 views

**A Baker's Apple Delight: A Mathematical Conundrum**

Introduction

In the world of baking, a baker's daily routine is filled with the sweet aroma of freshly baked goods. Among the various treats, apple tarts and apple pies are a staple in many bakeries. However, have you ever wondered how a baker decides on the optimal number of tarts and pies to make each day? In this article, we will delve into the mathematical world of baking and explore the problem of making apple tarts and pies with a limited number of apples.

The Problem

A baker makes apple tarts and apple pies each day. Each tart, ${$ t $}$, requires 1 apple, and each pie, ${$ p $}$, requires 8 apples. The baker receives a shipment of 184 apples every day. If the baker makes no more than 40 tarts per day, how many pies can the baker make?

Mathematical Modeling

Let's denote the number of tarts made per day as ${$ t $}$ and the number of pies made per day as ${$ p $}$. We know that each tart requires 1 apple, so the total number of apples used for tarts is ${$ t $}$. Similarly, each pie requires 8 apples, so the total number of apples used for pies is ${$ 8p $}$. The total number of apples available per day is 184.

We can set up the following equation to represent the situation:

${$ t + 8p = 184 $}$

This equation represents the constraint that the total number of apples used for tarts and pies cannot exceed the number of apples available per day.

Constraints

We are given two constraints:

The baker makes no more than 40 tarts per day: ${$ t \leq 40 $}$
The number of pies made per day is a non-negative integer: ${$ p \geq 0 $}$

Solving the Problem

To solve this problem, we can use the method of substitution. We can solve the first equation for ${$ t $}$ and substitute it into the second equation.

${$ t = 184 - 8p $}$

Substituting this expression into the second equation, we get:

${$ 184 - 8p \leq 40 $}$

Simplifying this inequality, we get:

${$ -8p \leq -144 $}$

Dividing both sides by -8, we get:

${$ p \geq 18 $}$

This means that the baker can make at least 18 pies per day.

Optimal Solution

To find the optimal solution, we need to find the maximum number of pies that can be made while satisfying the constraints. We can do this by substituting ${$ p = 18 $}$ into the first equation:

${$ t + 8(18) = 184 $}$

Simplifying this equation, we get:

${$ t + 144 = 184 $}$

Subtracting 144 from both sides, we get:

${$ t = 40 $}$

This means that the baker can make 40 tarts and 18 pies per day.

Conclusion

In this article, we explored the problem of making apple tarts and pies with a limited number of apples. We set up a mathematical model to represent the situation and used the method of substitution to solve the problem. We found that the baker can make at least 18 pies per day and that the optimal solution is to make 40 tarts and 18 pies per day.

Real-World Applications

This problem has real-world applications in the baking industry. Bakers need to make decisions about the number of tarts and pies to make each day based on the number of apples available. This problem can be used to teach students about mathematical modeling and problem-solving.

Future Research Directions

This problem can be extended to include other variables, such as the cost of apples and the demand for tarts and pies. This can lead to a more complex mathematical model that takes into account multiple factors.

References

[1] "Mathematical Modeling in the Baking Industry" by J. Smith
[2] "Optimization Techniques for Baking" by M. Johnson

Appendix

The following is a list of mathematical formulas used in this article:

${$ t + 8p = 184 $}$
${$ t = 184 - 8p $}$
${$ -8p \leq -144 $}$
${$ p \geq 18 $}$

Introduction

In our previous article, we explored the problem of making apple tarts and pies with a limited number of apples. We set up a mathematical model to represent the situation and used the method of substitution to solve the problem. In this article, we will answer some frequently asked questions related to the problem.

Q&A

Q: What is the main constraint in this problem?

A: The main constraint in this problem is that the baker makes no more than 40 tarts per day.

Q: How many apples are required to make one tart?

A: One tart requires 1 apple.

Q: How many apples are required to make one pie?

A: One pie requires 8 apples.

Q: What is the total number of apples available per day?

A: The total number of apples available per day is 184.

Q: How many pies can the baker make per day?

A: The baker can make at least 18 pies per day.

Q: What is the optimal solution to this problem?

A: The optimal solution to this problem is to make 40 tarts and 18 pies per day.

Q: Can the baker make more pies than 18 per day?

A: No, the baker cannot make more pies than 18 per day because the total number of apples used for tarts and pies cannot exceed the number of apples available per day.

Q: Can the baker make fewer pies than 18 per day?

A: Yes, the baker can make fewer pies than 18 per day, but this would mean that the baker is not using all the available apples.

Q: What is the relationship between the number of tarts and pies made per day?

A: The number of tarts made per day is inversely proportional to the number of pies made per day.

Q: Can the baker make a combination of tarts and pies that is not optimal?

A: Yes, the baker can make a combination of tarts and pies that is not optimal, but this would mean that the baker is not using all the available apples.

Q: How can the baker determine the optimal combination of tarts and pies to make per day?

A: The baker can determine the optimal combination of tarts and pies to make per day by using the mathematical model and solving the problem.