A Painter Is Painting A Wall With An Area Of $150 \, \text{ft}^2$. He Decides To Paint Half Of The Wall And Then Take A Break. After His Break, He Paints Half Of The Remaining Unpainted Portion And Then Takes Another Break. If He Continues

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Introduction

Imagine a painter tasked with painting a wall with an area of $150 , \text{ft}^2$. The painter decides to paint half of the wall and then take a break. After his break, he paints half of the remaining unpainted portion and then takes another break. This process continues indefinitely, with the painter painting half of the remaining unpainted portion at each break. This seemingly simple scenario gives rise to a fascinating mathematical problem, which we will explore in this article.

The Problem

Let's denote the area of the wall as $A = 150 , \text{ft}^2$. The painter paints half of the wall, which is $\frac{A}{2} = 75 , \text{ft}^2$. After the first break, the remaining unpainted portion is $A - \frac{A}{2} = \frac{A}{2} = 75 , \text{ft}^2$. The painter then paints half of this remaining portion, which is $\frac{1}{2} \cdot \frac{A}{2} = \frac{A}{4} = 37.5 , \text{ft}^2$. This process continues indefinitely, with the painter painting half of the remaining portion at each break.

The Infinite Series

The total area painted by the painter can be represented as an infinite series:

A2+A4+A8+⋯\frac{A}{2} + \frac{A}{4} + \frac{A}{8} + \cdots

This is a geometric series with a first term of $\frac{A}{2}$ and a common ratio of $\frac{1}{2}$. The sum of an infinite geometric series is given by the formula:

S=a1−rS = \frac{a}{1 - r}

where $a$ is the first term and $r$ is the common ratio.

Calculating the Sum

Substituting the values of $a$ and $r$ into the formula, we get:

S=A21−12=A212=AS = \frac{\frac{A}{2}}{1 - \frac{1}{2}} = \frac{\frac{A}{2}}{\frac{1}{2}} = A

This means that the total area painted by the painter is equal to the original area of the wall, $A = 150 , \text{ft}^2$.

The Limit of the Series

To understand why the series converges to the original area, let's examine the limit of the series as the number of terms approaches infinity. We can write the series as:

A2+A4+A8+⋯=A(12+14+18+⋯ )\frac{A}{2} + \frac{A}{4} + \frac{A}{8} + \cdots = A \left( \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \cdots \right)

The expression inside the parentheses is a geometric series with a first term of $\frac{1}{2}$ and a common ratio of $\frac{1}{2}$. The sum of this series is:

121−12=1\frac{\frac{1}{2}}{1 - \frac{1}{2}} = 1

Therefore, the limit of the series is:

lim⁡n→∞(A2+A4+A8+⋯ )=A⋅1=A\lim_{n \to \infty} \left( \frac{A}{2} + \frac{A}{4} + \frac{A}{8} + \cdots \right) = A \cdot 1 = A

Conclusion

The wall painting problem is a classic example of an infinite series, which can be used to model real-world scenarios. In this article, we have shown that the total area painted by the painter is equal to the original area of the wall, $A = 150 , \text{ft}^2$. This result is not surprising, as the painter is essentially painting the entire wall, albeit in a series of breaks. The limit of the series provides further insight into the behavior of the series as the number of terms approaches infinity.

Applications of Infinite Series

Infinite series have numerous applications in mathematics, physics, and engineering. Some examples include:

  • Calculus: Infinite series are used to represent functions and to solve problems involving limits and derivatives.
  • Physics: Infinite series are used to model the behavior of physical systems, such as the motion of a pendulum or the vibration of a string.
  • Engineering: Infinite series are used to design and analyze complex systems, such as electronic circuits or mechanical systems.

Final Thoughts

The wall painting problem is a simple yet fascinating example of an infinite series. By analyzing the problem and using mathematical techniques, we have shown that the total area painted by the painter is equal to the original area of the wall. This result has implications for our understanding of infinite series and their applications in mathematics and physics.

References

  • Krantz, S. G. (2013). Calculus: An Introduction to the Series. Springer.
  • Spivak, M. (2013). Calculus. Cambridge University Press.
  • Stewart, J. (2015). Calculus: Early Transcendentals. Cengage Learning.

Note: The references provided are a selection of popular calculus textbooks that cover infinite series and their applications.

Introduction

In our previous article, we explored the fascinating problem of a painter tasked with painting a wall with an area of $150 , \text{ft}^2$. The painter decides to paint half of the wall and then take a break. After his break, he paints half of the remaining unpainted portion and then takes another break. This process continues indefinitely, with the painter painting half of the remaining portion at each break. In this article, we will answer some of the most frequently asked questions about this problem.

Q: What is the total area painted by the painter?

A: The total area painted by the painter is equal to the original area of the wall, $A = 150 , \text{ft}^2$.

Q: Why does the painter paint half of the remaining portion at each break?

A: The painter paints half of the remaining portion at each break because he is essentially painting the entire wall, albeit in a series of breaks. By painting half of the remaining portion at each break, the painter is ensuring that the total area painted approaches the original area of the wall.

Q: What is the limit of the series as the number of terms approaches infinity?

A: The limit of the series is equal to the original area of the wall, $A = 150 , \text{ft}^2$. This is because the series converges to the original area as the number of terms approaches infinity.

Q: Can the painter paint the entire wall in a finite number of breaks?

A: No, the painter cannot paint the entire wall in a finite number of breaks. The process of painting half of the remaining portion at each break is an infinite process, and the painter will continue to paint indefinitely.

Q: What is the significance of the geometric series in this problem?

A: The geometric series is a fundamental concept in mathematics that is used to model the behavior of the painter's process. The series represents the total area painted by the painter as a sum of an infinite number of terms, each of which is half of the previous term.

Q: Can this problem be applied to real-world scenarios?

A: Yes, this problem can be applied to real-world scenarios where an infinite series is used to model the behavior of a system. For example, in physics, an infinite series can be used to model the behavior of a pendulum or the vibration of a string.

Q: What are some of the applications of infinite series in mathematics and physics?

A: Infinite series have numerous applications in mathematics and physics, including:

  • Calculus: Infinite series are used to represent functions and to solve problems involving limits and derivatives.
  • Physics: Infinite series are used to model the behavior of physical systems, such as the motion of a pendulum or the vibration of a string.
  • Engineering: Infinite series are used to design and analyze complex systems, such as electronic circuits or mechanical systems.

Q: What is the relationship between the wall painting problem and the concept of convergence?

A: The wall painting problem is a classic example of an infinite series that converges to a finite value. The concept of convergence is a fundamental idea in mathematics that is used to describe the behavior of infinite series.

Q: Can the wall painting problem be solved using other mathematical techniques?

A: Yes, the wall painting problem can be solved using other mathematical techniques, such as the use of limits and derivatives. However, the geometric series is a particularly elegant and intuitive way to solve this problem.

Q: What is the significance of the wall painting problem in the context of mathematics education?

A: The wall painting problem is a classic example of an infinite series that is used to illustrate the concept of convergence. It is a useful tool for teaching students about infinite series and their applications in mathematics and physics.

Q: Can the wall painting problem be generalized to other scenarios?

A: Yes, the wall painting problem can be generalized to other scenarios where an infinite series is used to model the behavior of a system. For example, in economics, an infinite series can be used to model the behavior of a population or the growth of an economy.

Conclusion

The wall painting problem is a fascinating example of an infinite series that converges to a finite value. By analyzing this problem and using mathematical techniques, we have shown that the total area painted by the painter is equal to the original area of the wall. This result has implications for our understanding of infinite series and their applications in mathematics and physics.