The Table Below Shows The Number Of Hours That Men Take To Complete A Particular Task.3.7.1 Complete The Table:$[ \begin{tabular}{|l|l|l|l|l|l|l|} \hline Number Of Men To Do Task & 1 & 2 & 4 & 6 & & 12 \ \hline Hours To Complete A Task & 8 & 4 &

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The Table of Men's Task Completion Times: A Mathematical Analysis

The table below presents the number of hours that men take to complete a particular task, with varying numbers of men involved. This data can be used to analyze the relationship between the number of men and the time taken to complete the task. In this article, we will complete the table and discuss the mathematical concepts underlying this problem.

Number of Men to do task 1 2 4 6 8 12
Hours to complete a task 8 4

To complete the table, we need to find the time taken by 8 men and 12 men to complete the task. Let's assume that the time taken by 1 man to complete the task is x hours. Then, the time taken by 2 men to complete the task is 2x/2 = x hours, which is given as 4 hours. Therefore, x = 4 hours.

Now, we can find the time taken by 8 men and 12 men to complete the task. The time taken by 8 men is 8x/8 = x hours, which is equal to 4 hours. Similarly, the time taken by 12 men is 12x/12 = x hours, which is also equal to 4 hours.

However, this is not the case. The time taken by 8 men and 12 men to complete the task is not equal to 4 hours. We need to find the correct values.

Let's analyze the situation. When 2 men are working together, they can complete the task in 4 hours. This means that the combined work rate of 2 men is 1/4 of the task per hour. When 4 men are working together, they can complete the task in 2 hours. This means that the combined work rate of 4 men is 1/2 of the task per hour.

Using this information, we can find the time taken by 8 men and 12 men to complete the task. The combined work rate of 8 men is 4/2 = 2 of the task per hour, which means they can complete the task in 2 hours. Similarly, the combined work rate of 12 men is 6/2 = 3 of the task per hour, which means they can complete the task in 2 hours.

Therefore, the completed table is:

Number of Men to do task 1 2 4 6 8 12
Hours to complete a task 8 4 2 2 2 2

The table shows that the time taken by men to complete the task decreases as the number of men increases. This is because the combined work rate of men increases as the number of men increases.

The relationship between the number of men and the time taken to complete the task can be represented by the equation:

T = k / n

where T is the time taken to complete the task, k is a constant, and n is the number of men.

We can find the value of k by using the data from the table. When n = 2, T = 4, so k = 8. Therefore, the equation becomes:

T = 8 / n

This equation shows that the time taken to complete the task decreases as the number of men increases.

In conclusion, we have completed the table and analyzed the relationship between the number of men and the time taken to complete the task. The table shows that the time taken by men to complete the task decreases as the number of men increases. The relationship between the number of men and the time taken to complete the task can be represented by the equation T = 8 / n.

The problem involves the concept of work rate, which is the rate at which a person or a group of people can complete a task. The work rate of a person or a group of people is measured in terms of the fraction of the task that they can complete per hour.

The problem also involves the concept of inverse proportionality, which is a relationship between two variables where one variable increases as the other variable decreases. In this case, the time taken to complete the task decreases as the number of men increases.

The concept of work rate and inverse proportionality has many real-world applications. For example, in manufacturing, the work rate of a machine or a group of machines can be used to determine the time taken to complete a production run. In logistics, the work rate of a team of people can be used to determine the time taken to complete a delivery.

In conclusion, the table of men's task completion times is a useful tool for analyzing the relationship between the number of men and the time taken to complete a task. The mathematical concepts underlying this problem, such as work rate and inverse proportionality, have many real-world applications.
Frequently Asked Questions (FAQs) about the Table of Men's Task Completion Times

A: The table shows that the time taken by men to complete the task decreases as the number of men increases. This is because the combined work rate of men increases as the number of men increases.

A: We can represent the relationship between the number of men and the time taken to complete the task using the equation T = 8 / n, where T is the time taken to complete the task, and n is the number of men.

A: The value of the constant k is 8, which can be found by using the data from the table. When n = 2, T = 4, so k = 8.

A: The work rate of a person or a group of people is the rate at which they can complete a task. It is measured in terms of the fraction of the task that they can complete per hour.

A: Inverse proportionality is a relationship between two variables where one variable increases as the other variable decreases. In this case, the time taken to complete the task decreases as the number of men increases.

A: Some real-world applications of the concept of work rate and inverse proportionality include manufacturing, logistics, and project management. For example, in manufacturing, the work rate of a machine or a group of machines can be used to determine the time taken to complete a production run. In logistics, the work rate of a team of people can be used to determine the time taken to complete a delivery.

A: Yes, we can use the equation T = 8 / n to predict the time taken to complete a task with a different number of men. Simply plug in the value of n into the equation to find the predicted time taken to complete the task.

A: Some limitations of the equation T = 8 / n include the assumption that the work rate of each individual is constant, and that the task is a simple one that can be completed in a linear fashion. In reality, tasks may be more complex and may involve multiple stages, which can affect the work rate and the time taken to complete the task.

A: Yes, we can use the equation T = 8 / n to compare the efficiency of different teams or individuals. By comparing the time taken to complete a task with a different number of men, we can determine which team or individual is more efficient.

A: Some potential applications of the equation T = 8 / n in real-world scenarios include:

  • Determining the time taken to complete a project with a team of people
  • Comparing the efficiency of different teams or individuals
  • Predicting the time taken to complete a task with a different number of men
  • Optimizing the number of men required to complete a task in a given time frame

In conclusion, the equation T = 8 / n provides a useful tool for analyzing the relationship between the number of men and the time taken to complete a task. By understanding the concept of work rate and inverse proportionality, we can use this equation to predict the time taken to complete a task with a different number of men, and to compare the efficiency of different teams or individuals.