Zahra's Knitting Project Using Inequalities To Maximize Craft Fair Output

Jul 24, 2025 by ADMIN 74 views

Zahra's Craft Fair Knitting Project: Maximizing Scarf and Hat Production

Hey guys! Let's dive into a cool math problem that involves Zahra, who's getting ready for a craft fair. She loves knitting and wants to make some awesome scarves and hats to sell. But, she has a limited amount of yarn and a goal to make a certain number of items. This is where math, specifically a system of inequalities, comes to the rescue! Let's break down how we can use inequalities to help Zahra figure out the best way to use her resources and meet her goals. Think of this as a real-life optimization puzzle – super relevant and fun!

Understanding Zahra's Knitting Dilemma

Yarn Inventory: Zahra has a total of 2,640 yards of yarn. This is her main resource constraint. She can't use more yarn than she has!

Yarn Usage: Each scarf requires 200 yards of yarn, and each hat needs 150 yards. This tells us how much of her resource each item consumes.

Minimum Items: Zahra wants to knit at least 15 items in total. This is her production goal – she needs to have enough inventory to make a good impression at the craft fair.

Setting up the Inequalities

To represent this situation mathematically, we'll use a system of inequalities. Inequalities are perfect for situations where we have constraints (like limited yarn) and goals (like a minimum number of items). Let's define our variables:

Let x represent the number of scarves Zahra knits.
Let y represent the number of hats Zahra knits.

Now, we can translate the information above into mathematical inequalities:

Yarn Constraint: 200x + 150y ≤ 2640
- This inequality represents the total yarn used. The amount of yarn used for scarves (200x) plus the amount of yarn used for hats (150y) must be less than or equal to the total yarn Zahra has (2640 yards).
Minimum Items Constraint: x + y ≥ 15
- This inequality represents the minimum number of items Zahra wants to knit. The total number of scarves (x) plus the total number of hats (y) must be greater than or equal to 15.
Non-Negativity Constraints: x ≥ 0 and y ≥ 0
- These inequalities are important because Zahra can't knit a negative number of scarves or hats. These are often called non-negativity constraints and are common in real-world optimization problems.

Deep Dive into the Yarn Constraint: 200x + 150y ≤ 2640

This inequality is the cornerstone of our problem, guys. It dictates how Zahra's limited yarn supply constrains her production. The left side, 200x + 150y, represents the total yarn consumption. Each scarf eats up 200 yards, and each hat uses 150 yards. So, if Zahra knits, say, 5 scarves and 4 hats, the total yarn used would be (200 * 5) + (150 * 4) = 1000 + 600 = 1600 yards. The ≤ sign tells us that this total consumption must be less than or equal to 2640 yards, which is Zahra's total yarn stash. Think of it like this: Zahra can't spend more money than she has in her bank account. Similarly, she can't use more yarn than she has available. This inequality ensures that Zahra stays within her yarn budget.

To further illustrate, let's consider some scenarios. What if Zahra decides to knit only scarves? In that case, y = 0, and the inequality becomes 200x ≤ 2640. Dividing both sides by 200, we get x ≤ 13.2. Since Zahra can't knit a fraction of a scarf, she can knit a maximum of 13 scarves if she makes no hats. On the other hand, if Zahra decides to knit only hats, then x = 0, and the inequality becomes 150y ≤ 2640. Dividing both sides by 150, we get y ≤ 17.6. So, she can knit a maximum of 17 hats if she makes no scarves. These extreme scenarios give us a sense of the boundaries imposed by the yarn constraint. But what about combinations of scarves and hats? That's where the power of the inequality shines, allowing us to explore a whole range of possibilities.

Exploring the Minimum Items Constraint: x + y ≥ 15

Now, let's shine a spotlight on the second inequality: x + y ≥ 15. This inequality captures Zahra's ambition to have a decent inventory at the craft fair. She wants to showcase a variety of items, so she's set a minimum goal of knitting 15 pieces. The left side, x + y, is simply the total number of items Zahra knits, whether they are scarves or hats. The ≥ sign means this total must be greater than or equal to 15. Zahra wants to make at least 15 items, but she's happy to make more if she can!

To get a better feel for this inequality, let's play with some numbers again. Suppose Zahra knits 10 scarves (x = 10). How many hats does she need to knit to meet her minimum items goal? Plugging x = 10 into the inequality, we get 10 + y ≥ 15. Subtracting 10 from both sides, we find y ≥ 5. So, Zahra needs to knit at least 5 hats to reach her goal of 15 items. If she knits 12 hats, she'll have a total of 10 + 12 = 22 items, which comfortably exceeds her minimum. What if she only knits 7 scarves? Then, 7 + y ≥ 15, which means y ≥ 8. She'd need to knit at least 8 hats. This inequality acts as a floor, ensuring Zahra has enough inventory to make her booth attractive at the craft fair. It's not just about using up all the yarn; it's also about meeting a minimum production target.

Don't Forget the Non-Negativity Constraints: x ≥ 0 and y ≥ 0

These inequalities might seem obvious, but they're absolutely crucial in the context of real-world problems. x ≥ 0 simply means that Zahra can't knit a negative number of scarves. It's impossible to produce -3 scarves, right? Similarly, y ≥ 0 means she can't knit a negative number of hats. These constraints reflect the physical reality of the situation. We're dealing with tangible objects – scarves and hats – and we can't have a negative quantity of them. In the mathematical world, these constraints restrict our solutions to the first quadrant of the coordinate plane (where both x and y are positive). Without these constraints, our solution space would be much larger and include unrealistic scenarios. Think of them as the guardrails that keep our solutions grounded in the real world.

Solving the System of Inequalities (The Next Step!)

Now that we've set up the system of inequalities, the next step would be to solve it. This involves finding the values of x and y (the number of scarves and hats) that satisfy all the inequalities simultaneously. Graphically, this means finding the region where the shaded areas of all the inequalities overlap. This overlapping region represents the set of all possible solutions. We could also use algebraic methods, like substitution or elimination, to find solutions. The ultimate goal is often to optimize something – maybe Zahra wants to maximize her profit, or minimize the amount of yarn left over. This optimization usually involves finding a specific solution within the feasible region that gives the best result according to some objective function. But for now, we've successfully translated Zahra's knitting challenge into a mathematical framework using a system of inequalities. Pretty cool, huh?

The Importance of Systems of Inequalities in Real Life

This knitting scenario might seem like a specific example, guys, but systems of inequalities are used everywhere in real life! They're a powerful tool for modeling situations with constraints and goals. Think about:

Business: A company might use inequalities to determine the optimal production levels of different products, given constraints on resources like raw materials, labor, and machine time. They might want to maximize profit while staying within budget.
Nutrition: A dietitian might use inequalities to create a meal plan that meets certain nutritional requirements (like minimum protein and vitamin intake) while staying within a calorie limit.
Logistics: A shipping company might use inequalities to optimize delivery routes, considering constraints like vehicle capacity, delivery time windows, and fuel costs.
Finance: Investors might use inequalities to create a portfolio that balances risk and return, given constraints on investment amounts and diversification requirements.

In all these situations, inequalities help us find the best possible solution within a set of limitations. They're a key tool for decision-making and optimization. So, understanding how to set up and solve systems of inequalities is a valuable skill that can be applied in many different fields. Whether it's Zahra's knitting project or a multi-million dollar business decision, the underlying mathematical principles are the same.

Conclusion: Zahra's Knitting Success

So, there you have it! We've helped Zahra translate her knitting goals and limitations into a system of inequalities. This is a crucial first step in figuring out how many scarves and hats she should knit to make the most of her yarn and have a successful craft fair. Remember, the inequalities we set up – 200x + 150y ≤ 2640, x + y ≥ 15, x ≥ 0, and y ≥ 0 – represent Zahra's yarn constraint, minimum item goal, and the non-negativity of her production. These inequalities create a mathematical model of the real-world situation, allowing us to use powerful mathematical tools to find the best solution. By understanding these concepts, you're not just solving math problems; you're learning how to make informed decisions in a variety of situations. And who knows, maybe you'll even use this knowledge to start your own craft fair business someday!