The Time Taken, $t$, For Passengers To Be Checked-in For A Flight Is Inversely Proportional To The Square Of The Number Of Staff, $s$, Working.It Takes 30 Minutes For Passengers To Be Checked-in When 10 Staff Are Working.Given:

by ADMIN 232 views

The Time Taken for Passengers to be Checked-in: An Inverse Proportionality Problem

In this article, we will explore the concept of inverse proportionality and its application to a real-world scenario. We will examine the relationship between the time taken for passengers to be checked-in and the number of staff working. The problem states that the time taken, $t$, for passengers to be checked-in is inversely proportional to the square of the number of staff, $s$, working. This means that as the number of staff increases, the time taken decreases, and the relationship between the two variables can be represented by the equation $t = \frac{k}{s^2}$, where $k$ is a constant of proportionality.

Understanding Inverse Proportionality

Inverse proportionality is a relationship between two variables where one variable increases as the other decreases, and vice versa. In this case, the time taken for passengers to be checked-in decreases as the number of staff increases. This type of relationship can be represented by an inverse proportionality equation, which takes the form $y = \frac{k}{x}$, where $y$ is the dependent variable, $x$ is the independent variable, and $k$ is a constant of proportionality.

The Given Information

We are given that it takes 30 minutes for passengers to be checked-in when 10 staff are working. This information can be used to find the constant of proportionality, $k$. We can substitute the given values into the equation $t = \frac{k}{s^2}$ and solve for $k$.

Finding the Constant of Proportionality

We are given that $t = 30$ minutes and $s = 10$. Substituting these values into the equation $t = \frac{k}{s^2}$, we get:

30=k10230 = \frac{k}{10^2}

Simplifying the equation, we get:

30=k10030 = \frac{k}{100}

Multiplying both sides of the equation by 100, we get:

3000=k3000 = k

Therefore, the constant of proportionality is $k = 3000$.

The Equation of Inverse Proportionality

Now that we have found the constant of proportionality, we can write the equation of inverse proportionality:

t=3000s2t = \frac{3000}{s^2}

This equation represents the relationship between the time taken for passengers to be checked-in and the number of staff working.

Solving for the Number of Staff

We can use the equation of inverse proportionality to solve for the number of staff working when the time taken is 20 minutes. We can substitute the given value of $t$ into the equation and solve for $s$.

20=3000s220 = \frac{3000}{s^2}

Multiplying both sides of the equation by $s^2$, we get:

20s2=300020s^2 = 3000

Dividing both sides of the equation by 20, we get:

s2=150s^2 = 150

Taking the square root of both sides of the equation, we get:

s=150s = \sqrt{150}

Simplifying the expression, we get:

s=12.25s = 12.25

Therefore, the number of staff working when the time taken is 20 minutes is approximately 12.25.

In this article, we have explored the concept of inverse proportionality and its application to a real-world scenario. We have examined the relationship between the time taken for passengers to be checked-in and the number of staff working. We have found the constant of proportionality and written the equation of inverse proportionality. We have also solved for the number of staff working when the time taken is 20 minutes. This problem demonstrates the importance of understanding inverse proportionality and its application to real-world scenarios.

For further reading on inverse proportionality, we recommend the following resources:

  • Khan Academy: Inverse Proportionality
  • Math Is Fun: Inverse Proportion
  • Wolfram MathWorld: Inverse Proportion

Q: What is inverse proportionality?

A: Inverse proportionality is a relationship between two variables where one variable increases as the other decreases, and vice versa. In the context of checking-in passengers, the time taken for passengers to be checked-in decreases as the number of staff increases.

Q: How is inverse proportionality represented mathematically?

A: Inverse proportionality can be represented mathematically by the equation $t = \frac{k}{s^2}$, where $t$ is the time taken for passengers to be checked-in, $s$ is the number of staff working, and $k$ is a constant of proportionality.

Q: What is the constant of proportionality, $k$?

A: The constant of proportionality, $k$, is a value that represents the relationship between the time taken for passengers to be checked-in and the number of staff working. In this problem, we found that $k = 3000$.

Q: How do I use the equation of inverse proportionality to solve for the number of staff working?

A: To solve for the number of staff working, you can substitute the given value of $t$ into the equation $t = \frac{k}{s^2}$ and solve for $s$. For example, if the time taken is 20 minutes, you can substitute $t = 20$ into the equation and solve for $s$.

Q: What is the relationship between the number of staff working and the time taken for passengers to be checked-in?

A: The number of staff working and the time taken for passengers to be checked-in are inversely proportional. This means that as the number of staff increases, the time taken decreases, and vice versa.

Q: How can I apply the concept of inverse proportionality to real-world scenarios?

A: The concept of inverse proportionality can be applied to many real-world scenarios, such as:

  • The relationship between the number of workers and the time taken to complete a task
  • The relationship between the number of customers and the time taken to serve them
  • The relationship between the number of staff and the time taken to check-in passengers

Q: What are some common mistakes to avoid when working with inverse proportionality?

A: Some common mistakes to avoid when working with inverse proportionality include:

  • Failing to identify the constant of proportionality, $k$
  • Failing to use the correct equation of inverse proportionality
  • Failing to solve for the correct variable (in this case, the number of staff working)

Q: How can I practice working with inverse proportionality?

A: You can practice working with inverse proportionality by:

  • Solving problems that involve inverse proportionality
  • Creating your own problems that involve inverse proportionality
  • Using online resources, such as Khan Academy or Math Is Fun, to practice working with inverse proportionality.

In this article, we have answered some frequently asked questions about inverse proportionality and checking-in passengers. We have explored the concept of inverse proportionality, its mathematical representation, and its application to real-world scenarios. We have also provided some common mistakes to avoid and some tips for practicing working with inverse proportionality.