Optimizing Workforce for Efficient Spraying of Insecticides: A Mathematical Analysis
In the context of agricultural work, particularly when dealing with the spraying of insecticides on fruit trees, there are specific parameters that must be carefully managed to achieve optimal efficiency. One such parameter involves determining the number of workers required to complete a task within a specified time frame. This article explores the principles of work-rate and man hours to find the most efficient approach for spraying insecticides over a given area.
Understanding the Initial Scenario
Consider the following scenario: ten men take 12 hours to spray insecticides on a group of fruit trees spread over 40 hectares. The objective is to determine how many men will be required to complete the same task, but over a smaller area, in a shorter time frame.
First, let's calculate the total man hours required to complete the task in the initial scenario:
Man hours Number of men times; Time (in hours) 10 men times; 12 hours 120 man hours
Using Proportional Relationships
The relationship between the time taken, the amount of land that needs to be sprayed, and the number of men can be described using a proportional relationship. The time T required is directly proportional to the land extent A and inversely proportional to the number of men N working. This can be mathematically expressed as:
T propto; A/N
Or, in terms of a constant K, the equation becomes:
T K times; A/N
Where K is a constant that does not change during the given task.
Solving for the Required Manpower
We need to find the number of men required for a different scenario where the land extent is 32 hectares and the time required is 8 hours. We will use the initial conditions to solve for N in the new scenario.
Step-by-Step Calculation
Calculate the constant K using the initial conditions:120 man hours K times; 40 hectares / 10 men
K 120 times; 10 / 40 30
Use K to solve for the new number of men:T times; A/N K
8 hours times; 32 hectares / N 30
256 / N 30
N 256 / 30 8.53
Rounding up to the nearest whole number:N 12 men
Hence, the task will take 12 men to spray a 32-hectare area in 8 hours.
Conclusion: By understanding the principles of proportionality between man hours, time, and the area to be covered, we can efficiently plan and allocate resources for agricultural tasks. The analysis demonstrates that with proper planning, the number of workers required can be determined accurately, leading to optimized resource management and productivity.