Width and Length Inequality

Oct 13, 2025 by miodeya.com 146 views

Introduction

In this article, we will delve into a mathematical problem presented by Charlene, a crafty individual who is knitting a baby blanket. Charlene's primary objective is to create a blanket with a width, denoted as $w$ , that is at least half its length, denoted as $l$ . Additionally, she wants to ensure that the perimeter of the blanket does not exceed 180 units, as she has enough yarn to put a fringe around it. This problem requires us to apply mathematical concepts, such as inequalities and perimeter calculations, to find the possible dimensions of the blanket.

Problem Formulation

Let's denote the width of the blanket as $w$ and its length as $l$ . Charlene's requirements can be expressed as two inequalities:

Width and Length Inequality: $w \geq \frac{1}{2}l$
Perimeter Constraint: $2w + 2l \leq 180$

These inequalities represent the conditions that Charlene wants to satisfy. We will use these inequalities to find the possible values of $w$ and $l$ that meet her requirements.

Solving the Inequality

To solve the inequality $w \geq \frac{1}{2}l$ , we can multiply both sides by 2 to eliminate the fraction:

$2w \geq l$

This inequality represents the condition that the width of the blanket must be at least half its length.

Perimeter Calculation

The perimeter of a rectangle, such as the baby blanket, is given by the formula:

$P = 2w + 2l$

We are given that the perimeter must not exceed 180 units, so we can write the inequality:

$2w + 2l \leq 180$

We can simplify this inequality by dividing both sides by 2:

$w + l \leq 90$

This inequality represents the condition that the sum of the width and length of the blanket must not exceed 90 units.

Combining the Inequalities

We have two inequalities that represent Charlene's requirements:

$2w \geq l$
$w + l \leq 90$

We can use these inequalities to find the possible values of $w$ and $l$ that meet Charlene's requirements.

Graphical Representation

To visualize the solution to the inequalities, we can graph the lines $2w = l$ and $w + l = 90$ on a coordinate plane. The region that satisfies both inequalities will represent the possible values of $w$ and $l$ .

Graph of the Inequalities

The graph of the inequality $2w = l$ is a line with a slope of 2 and a y-intercept of 0. The graph of the inequality $w + l = 90$ is a line with a slope of -1 and a y-intercept of 90.

Region of Possible Values

The region that satisfies both inequalities is the area below the line $2w = l$ and to the left of the line $w + l = 90$ . This region represents the possible values of $w$ and $l$ that meet Charlene's requirements.

Finding the Possible Values

To find the possible values of $w$ and $l$ , we can use the inequalities to set up a system of equations. We can solve this system of equations to find the values of $w$ and $l$ that satisfy both inequalities.

System of Equations

We can set up the following system of equations:

$2w = l$
$w + l = 90$

We can solve this system of equations by substitution or elimination.

Solving the System of Equations

We can solve the system of equations by substitution. We can substitute the expression for $l$ from the first equation into the second equation:

$w + 2w = 90$

Combine like terms:

$3w = 90$

Divide both sides by 3:

$w = 30$

Now that we have found the value of $w$ , we can substitute it into one of the original equations to find the value of $l$ . We can use the first equation:

$l = 2w$

Substitute the value of $w$ :

$l = 2(30)$

$l = 60$

Therefore, the possible values of $w$ and $l$ are $w = 30$ and $l = 60$ .

Conclusion

Frequently Asked Questions

In this article, we will address some of the most frequently asked questions related to Charlene's baby blanket conundrum.

Q: What is the width of the baby blanket?

A: The width of the baby blanket is denoted as $w$ . According to Charlene's requirements, the width must be at least half the length of the blanket.

Q: What is the length of the baby blanket?

A: The length of the baby blanket is denoted as $l$ . According to Charlene's requirements, the length must be such that the perimeter of the blanket does not exceed 180 units.

Q: How do I calculate the perimeter of the baby blanket?

A: The perimeter of a rectangle, such as the baby blanket, is given by the formula:

$P = 2w + 2l$

You can use this formula to calculate the perimeter of the baby blanket.

Q: What is the relationship between the width and length of the baby blanket?

A: According to Charlene's requirements, the width of the baby blanket must be at least half its length. This can be expressed as the inequality:

$w \geq \frac{1}{2}l$

Q: How do I find the possible values of $w$ and $l$ that meet Charlene's requirements?

A: To find the possible values of $w$ and $l$ , you can use the inequalities to set up a system of equations. You can solve this system of equations to find the values of $w$ and $l$ that satisfy both inequalities.

Q: What are the possible values of $w$ and $l$ that meet Charlene's requirements?

A: The possible values of $w$ and $l$ that meet Charlene's requirements are $w = 30$ and $l = 60$ .

Q: Why is it important to consider the perimeter of the baby blanket?

A: It is essential to consider the perimeter of the baby blanket because Charlene wants to put a fringe around it, and she has enough yarn to do so only if the perimeter does not exceed 180 units.

Q: Can I use the same method to solve other problems involving inequalities and perimeter calculations?

A: Yes, you can use the same method to solve other problems involving inequalities and perimeter calculations. The key is to understand the relationships between the variables and to use the inequalities to set up a system of equations.

Q: What are some real-world applications of the concepts learned in this article?

A: The concepts learned in this article have real-world applications in various fields, such as engineering, architecture, and design. For example, you can use these concepts to design buildings, bridges, or other structures that meet specific requirements.

Conclusion

In this article, we have addressed some of the most frequently asked questions related to Charlene's baby blanket conundrum. We have provided detailed explanations and examples to help you understand the concepts and apply them to real-world problems.