It Takes Dimitri 9 Minutes To Make A Simple Bracelet And 20 Minutes To Make A Deluxe Bracelet. He Has Been Making Bracelets For Longer Than 120 Minutes. If X X X Represents The Number Of Simple Bracelets That He Has Made And Y Y Y

Introduction

In this article, we will explore a problem involving Dimitri, a skilled craftsman who makes bracelets. We will use algebraic equations to represent the time it takes him to make simple and deluxe bracelets. The problem will help us understand how to solve systems of linear equations and how to use them to model real-world situations.

The Problem

Dimitri has been making bracelets for longer than 120 minutes. It takes him 9 minutes to make a simple bracelet and 20 minutes to make a deluxe bracelet. If $x$ represents the number of simple bracelets that he has made and $y$ represents the number of deluxe bracelets that he has made, we can write two equations to represent the total time he has spent making bracelets.

Equations

Let's start by writing the equations that represent the total time Dimitri has spent making bracelets. We know that the total time is the sum of the time spent making simple bracelets and the time spent making deluxe bracelets.

• The total time spent making simple bracelets is $9x$ minutes.
• The total time spent making deluxe bracelets is $20y$ minutes.

Since Dimitri has been making bracelets for longer than 120 minutes, we can write the following inequality:

$$9x + 20y > 120$$

We can also write an equation to represent the fact that the total number of bracelets is the sum of the number of simple bracelets and the number of deluxe bracelets:

$$x + y = n$$

where $n$ is the total number of bracelets.

Solving the System of Equations

To solve the system of equations, we can use the substitution method. We can solve the second equation for $x$ and substitute it into the first equation.

$$x = n - y$$

Substituting this expression for $x$ into the first equation, we get:

$$9(n - y) + 20y > 120$$

Expanding and simplifying the inequality, we get:

$$9n - 9y + 20y > 120$$

Combine like terms:

$$9n + 11y > 120$$

Now, we can solve the inequality for $y$.

$$11y > 120 - 9n$$

Divide both sides by 11:

$$y > \frac{120 - 9n}{11}$$

Since $y$ must be a non-negative integer, we can write:

$$y \geq \left\lceil \frac{120 - 9n}{11} \right\rceil$$

where $\lceil x \rceil$ is the ceiling function, which rounds $x$ up to the nearest integer.

Graphing the Inequality

To visualize the solution to the inequality, we can graph the region that satisfies the inequality. The graph will be a region in the $xy$-plane that includes all points that satisfy the inequality.

The graph will be a region that includes all points that satisfy the inequality $y \geq \left\lceil \frac{120 - 9n}{11} \right\rceil$. The region will be bounded by the line $y = \left\lceil \frac{120 - 9n}{11} \right\rceil$ and the $x$-axis.

Conclusion

In this article, we have explored a problem involving Dimitri, a skilled craftsman who makes bracelets. We have used algebraic equations to represent the time it takes him to make simple and deluxe bracelets. We have solved the system of equations and graphed the region that satisfies the inequality. The problem has helped us understand how to solve systems of linear equations and how to use them to model real-world situations.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Calculus: Early Transcendentals" by James Stewart

Glossary

• Simple bracelet: A type of bracelet that takes 9 minutes to make.
• Deluxe bracelet: A type of bracelet that takes 20 minutes to make.
• System of equations: A set of two or more equations that are solved simultaneously.
• Inequality: An expression that states that one quantity is greater than, less than, or equal to another quantity.
• Ceiling function: A function that rounds a number up to the nearest integer.
  Q&A: Dimitri's Bracelets ==========================

Introduction

In our previous article, we explored a problem involving Dimitri, a skilled craftsman who makes bracelets. We used algebraic equations to represent the time it takes him to make simple and deluxe bracelets. In this article, we will answer some frequently asked questions about Dimitri's bracelets.

Q: How long does it take Dimitri to make a simple bracelet?

A: It takes Dimitri 9 minutes to make a simple bracelet.

Q: How long does it take Dimitri to make a deluxe bracelet?

A: It takes Dimitri 20 minutes to make a deluxe bracelet.

Q: What is the total time spent making simple bracelets?

A: The total time spent making simple bracelets is $9x$ minutes, where $x$ is the number of simple bracelets made.

Q: What is the total time spent making deluxe bracelets?

A: The total time spent making deluxe bracelets is $20y$ minutes, where $y$ is the number of deluxe bracelets made.

Q: What is the inequality that represents the total time spent making bracelets?

A: The inequality that represents the total time spent making bracelets is $9x + 20y > 120$.

Q: How can we solve the system of equations?

A: We can solve the system of equations by using the substitution method. We can solve the second equation for $x$ and substitute it into the first equation.

Q: What is the graph of the inequality?

A: The graph of the inequality is a region in the $xy$-plane that includes all points that satisfy the inequality $y \geq \left\lceil \frac{120 - 9n}{11} \right\rceil$.

Q: What is the ceiling function?

A: The ceiling function is a function that rounds a number up to the nearest integer.

Q: How can we use the ceiling function to solve the inequality?

A: We can use the ceiling function to solve the inequality by rounding the expression $\frac{120 - 9n}{11}$ up to the nearest integer.

Q: What is the solution to the inequality?

A: The solution to the inequality is $y \geq \left\lceil \frac{120 - 9n}{11} \right\rceil$.

Q: How can we use the solution to the inequality to find the number of bracelets made?

A: We can use the solution to the inequality to find the number of bracelets made by plugging in different values of $n$ and finding the corresponding values of $y$.

Conclusion

In this article, we have answered some frequently asked questions about Dimitri's bracelets. We have used algebraic equations to represent the time it takes him to make simple and deluxe bracelets. We have solved the system of equations and graphed the region that satisfies the inequality. The problem has helped us understand how to solve systems of linear equations and how to use them to model real-world situations.

References

• [1] "Algebra and Trigonometry" by Michael Sullivan
• [2] "Mathematics for the Nonmathematician" by Morris Kline

Further Reading

• [1] "Linear Algebra and Its Applications" by Gilbert Strang
• [2] "Calculus: Early Transcendentals" by James Stewart

Glossary

• Simple bracelet: A type of bracelet that takes 9 minutes to make.
• Deluxe bracelet: A type of bracelet that takes 20 minutes to make.
• System of equations: A set of two or more equations that are solved simultaneously.
• Inequality: An expression that states that one quantity is greater than, less than, or equal to another quantity.
• Ceiling function: A function that rounds a number up to the nearest integer.