The Product Of Two Positive Numbers Is 1,800. One Number Is Twice The Other Number. The Equation To Determine The Two Numbers, With $x$ Representing The Value Of The Lesser Number, Is $x(2x)=1,800$.What Is The Value Of The Lesser

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Introduction

In mathematics, solving equations is a fundamental concept that helps us understand various relationships between variables. In this article, we will delve into a problem that involves finding two positive numbers whose product is 1,800, with one number being twice the other. We will use algebraic techniques to solve the equation and determine the value of the lesser number.

The Problem

The problem states that the product of two positive numbers is 1,800, and one number is twice the other. Let's represent the lesser number as $x$. Since the other number is twice the lesser number, we can represent it as $2x$. The equation to determine the two numbers is given by:

x(2x)=1,800x(2x) = 1,800

Solving the Equation

To solve the equation, we can start by expanding the left-hand side:

2x2=1,8002x^2 = 1,800

Next, we can divide both sides by 2 to isolate the term $x^2$:

x2=1,8002x^2 = \frac{1,800}{2}

x2=900x^2 = 900

Finding the Value of x

Now that we have the equation $x^2 = 900$, we can find the value of $x$ by taking the square root of both sides:

x=900x = \sqrt{900}

x=30x = 30

The Value of the Lesser Number

Since we have found the value of $x$, which represents the lesser number, we can conclude that the lesser number is 30.

Conclusion

In this article, we have explored a problem that involves finding two positive numbers whose product is 1,800, with one number being twice the other. We used algebraic techniques to solve the equation and determined the value of the lesser number. The value of the lesser number is 30.

Additional Insights

  • The problem can be solved using other algebraic techniques, such as factoring or using the quadratic formula.
  • The value of the greater number can be found by multiplying the value of the lesser number by 2.
  • The product of the two numbers can be verified by multiplying the value of the lesser number by the value of the greater number.

Real-World Applications

  • The problem can be applied to real-world scenarios, such as finding the dimensions of a rectangle with a fixed area.
  • The problem can be used to model real-world relationships, such as the relationship between the price and quantity of a product.

Mathematical Concepts

  • The problem involves the concept of quadratic equations and their solutions.
  • The problem requires the use of algebraic techniques, such as expanding and simplifying expressions.
  • The problem involves the concept of positive numbers and their properties.

Future Research Directions

  • Investigating the properties of quadratic equations and their solutions.
  • Exploring the applications of quadratic equations in real-world scenarios.
  • Developing new algebraic techniques for solving quadratic equations.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Appendix

  • The problem can be solved using other algebraic techniques, such as factoring or using the quadratic formula.
  • The value of the greater number can be found by multiplying the value of the lesser number by 2.
  • The product of the two numbers can be verified by multiplying the value of the lesser number by the value of the greater number.
    The Product of Two Positive Numbers: A Q&A Session =====================================================

Introduction

In our previous article, we explored a problem that involved finding two positive numbers whose product is 1,800, with one number being twice the other. We used algebraic techniques to solve the equation and determined the value of the lesser number. In this article, we will provide a Q&A session to help clarify any doubts and provide additional insights into the problem.

Q: What is the equation to determine the two numbers?

A: The equation to determine the two numbers is given by:

x(2x)=1,800x(2x) = 1,800

Q: How do we solve the equation?

A: To solve the equation, we can start by expanding the left-hand side:

2x2=1,8002x^2 = 1,800

Next, we can divide both sides by 2 to isolate the term $x^2$:

x2=1,8002x^2 = \frac{1,800}{2}

x2=900x^2 = 900

Q: How do we find the value of x?

A: Now that we have the equation $x^2 = 900$, we can find the value of $x$ by taking the square root of both sides:

x=900x = \sqrt{900}

x=30x = 30

Q: What is the value of the greater number?

A: Since the greater number is twice the lesser number, we can find its value by multiplying the value of the lesser number by 2:

2x=2(30)2x = 2(30)

2x=602x = 60

Q: How do we verify the product of the two numbers?

A: We can verify the product of the two numbers by multiplying the value of the lesser number by the value of the greater number:

x(2x)=30(60)x(2x) = 30(60)

x(2x)=1,800x(2x) = 1,800

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

A: The problem can be applied to real-world scenarios, such as finding the dimensions of a rectangle with a fixed area. It can also be used to model real-world relationships, such as the relationship between the price and quantity of a product.

Q: What are some mathematical concepts involved in this problem?

A: The problem involves the concept of quadratic equations and their solutions. It also requires the use of algebraic techniques, such as expanding and simplifying expressions. Additionally, it involves the concept of positive numbers and their properties.

Q: Can we solve this problem using other algebraic techniques?

A: Yes, we can solve this problem using other algebraic techniques, such as factoring or using the quadratic formula.

Q: What are some future research directions related to this problem?

A: Some potential future research directions include investigating the properties of quadratic equations and their solutions, exploring the applications of quadratic equations in real-world scenarios, and developing new algebraic techniques for solving quadratic equations.

Conclusion

In this Q&A session, we have provided additional insights and clarification on the problem of finding two positive numbers whose product is 1,800, with one number being twice the other. We hope that this article has been helpful in understanding the problem and its solutions.

References

  • [1] "Algebra and Trigonometry" by Michael Sullivan
  • [2] "Calculus" by Michael Spivak
  • [3] "Mathematics for Computer Science" by Eric Lehman and Tom Leighton

Appendix

  • The problem can be solved using other algebraic techniques, such as factoring or using the quadratic formula.
  • The value of the greater number can be found by multiplying the value of the lesser number by 2.
  • The product of the two numbers can be verified by multiplying the value of the lesser number by the value of the greater number.