Finding the Greatest Width

Oct 13, 2025 by miodeya.com 55 views

Introduction

When it comes to creating a mosaic, the possibilities are endless, but the constraints can be limiting. In this scenario, John has 48 square centimeter tiles at his disposal, and he wants to create a rectangular mosaic with a length that is 2 centimeters longer than the width. The question is, what is the greatest width John can achieve with the tiles he has? To answer this, we need to dive into the world of mathematics and explore the relationship between the tiles, the width, and the length of the mosaic.

Understanding the Problem

Let's break down the problem and understand what we're dealing with. John has 48 square centimeter tiles, and he wants to create a rectangular mosaic with a length that is 2 centimeters longer than the width. This means that if the width is w, the length will be w + 2. The area of the mosaic will be the product of the width and the length, which is w(w + 2).

The Equation

To find the greatest width John can achieve, we need to set up an equation that represents the relationship between the tiles, the width, and the length. Since John has 48 square centimeter tiles, the area of the mosaic must be equal to or less than 48. This gives us the following equation:

w(w + 2) ≤ 48

Solving the Equation

Now that we have the equation, let's solve for w. To do this, we can start by expanding the left-hand side of the equation:

w^2 + 2w ≤ 48

Next, we can rearrange the equation to get a quadratic equation in the form of ax^2 + bx + c = 0:

w^2 + 2w - 48 ≤ 0

To solve this quadratic equation, we can use the quadratic formula:

w = (-b ± √(b^2 - 4ac)) / 2a

In this case, a = 1, b = 2, and c = -48. Plugging these values into the formula, we get:

w = (-(2) ± √((2)^2 - 4(1)(-48))) / 2(1) w = (-2 ± √(4 + 192)) / 2 w = (-2 ± √196) / 2 w = (-2 ± 14) / 2

Finding the Greatest Width

Now that we have the solutions to the quadratic equation, we need to find the greatest width John can achieve. The two possible solutions are:

w = (-2 + 14) / 2 = 6 w = (-2 - 14) / 2 = -8

Since the width cannot be negative, we can discard the solution w = -8. This leaves us with the solution w = 6, which is the greatest width John can achieve with the tiles he has.

Conclusion

In conclusion, to find the greatest width John can achieve with the tiles he has, we need to set up an equation that represents the relationship between the tiles, the width, and the length of the mosaic. By solving the equation, we can find the maximum width, which in this case is 6 centimeters. This mathematical approach allows us to understand the constraints of the problem and find the optimal solution.

Additional Considerations

While we have found the greatest width John can achieve, there are some additional considerations to keep in mind. For example, the tiles may not be perfectly square, which could affect the area of the mosaic. Additionally, the length of the mosaic may not be exactly 2 centimeters longer than the width, which could also impact the area. However, for the purposes of this problem, we have assumed that the tiles are perfectly square and the length is exactly 2 centimeters longer than the width.

Real-World Applications

The mathematical approach used in this problem has real-world applications in various fields, such as architecture, engineering, and design. For example, when designing a building or a bridge, architects and engineers need to consider the constraints of the materials and the space available. By using mathematical equations and formulas, they can find the optimal solution that meets the requirements of the project.

Final Thoughts

Q: What is the main constraint in this problem?

A: The main constraint in this problem is the limited number of tiles John has, which is 48 square centimeter tiles.

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

A: The length of the mosaic is 2 centimeters longer than the width. If the width is w, the length will be w + 2.

Q: How do we find the greatest width John can achieve?

A: To find the greatest width John can achieve, we need to set up an equation that represents the relationship between the tiles, the width, and the length of the mosaic. We then solve the equation to find the maximum width.

Q: What is the equation we use to find the greatest width?

A: The equation we use to find the greatest width is w(w + 2) ≤ 48, where w is the width of the mosaic.

Q: How do we solve the equation w(w + 2) ≤ 48?

A: To solve the equation w(w + 2) ≤ 48, we can start by expanding the left-hand side of the equation: w^2 + 2w ≤ 48. We can then rearrange the equation to get a quadratic equation in the form of ax^2 + bx + c = 0.

Q: What is the quadratic formula we use to solve the equation?

A: The quadratic formula we use to solve the equation is w = (-b ± √(b^2 - 4ac)) / 2a, where a = 1, b = 2, and c = -48.

Q: What are the solutions to the quadratic equation?

A: The solutions to the quadratic equation are w = (-2 + 14) / 2 = 6 and w = (-2 - 14) / 2 = -8.

Q: Why do we discard the solution w = -8?

A: We discard the solution w = -8 because the width cannot be negative.

Q: What is the greatest width John can achieve?

A: The greatest width John can achieve is 6 centimeters.

Q: What are some additional considerations we need to keep in mind?

A: Some additional considerations we need to keep in mind are that the tiles may not be perfectly square, which could affect the area of the mosaic, and the length of the mosaic may not be exactly 2 centimeters longer than the width, which could also impact the area.

Q: What are some real-world applications of this mathematical problem?

A: Some real-world applications of this mathematical problem are in architecture, engineering, and design, where architects and engineers need to consider the constraints of the materials and the space available to find the optimal solution.

Q: What is the final answer to the problem?

A: The final answer to the problem is that the greatest width John can achieve with the tiles he has is 6 centimeters.