A government agency has a current staff of 5000, of whom 25% are women. Employees are quitting randomly at the rate of 100 per week. Replacements are being hired at the rate of 50 per week, with the requirement that half be women.
Interested in the rate of women coming in and out.
Set up a differential equation to the model the situation.
My first mistake I'm guessing was thinking about this like a mixture problem. The answer is dW/dt = 25 - 2W/(100 - t).
Can someone explain to me the thought process? I have no idea where the 't' comes from and why 100 is being subtracted by it. The 2W as well I have no idea about.
To model the situation of employees quitting and replacements being hired, we need to establish a differential equation that describes the rate of change of the number of women in the agency over time.
Let's break down the components of the equation:
- dW/dt: This represents the rate of change of the number of women in the agency with respect to time (t). It indicates how the number of women is changing over time.
- 25: This represents the initial percentage of women in the agency (25% or 0.25). This value stands for the starting point of the number of women.
- 2W: This term represents the rate at which women are leaving the agency. The coefficient of 2 can be explained by considering that 2 out of every 100 employees are quitting per week (0.25 * 100 = 25% are women, and 25% of the 100 employees who quit are women).
- (100 - t): This term represents the remaining number of weeks that employees will quit. As time progresses (week after week), the number of remaining weeks for employees to quit decreases. Since employees are quitting randomly at the rate of 100 per week, subtracting t (the number of weeks that have passed) from 100 accounts for this diminishing number of weeks.
Thus, the differential equation representing the rate of change of the number of women in the agency is:
dW/dt = 25 - 2W/(100 - t)
This equation describes how the number of women in the agency is affected by both the initial percentage of women and the rate at which employees are quitting, taking into account the diminishing number of remaining weeks for employees to quit.
To set up a differential equation to model the situation, we need to consider the rate at which employees are leaving the agency and the rate at which replacements are being hired.
Let's break it down step by step:
1. Employees leaving the agency:
According to the problem, employees are quitting randomly at the rate of 100 per week. This rate is constant over time.
2. Replacements being hired:
Replacements are being hired at the rate of 50 per week, with the requirement that half of them be women. Since we know that 25% of the current staff are women, this means that the number of women leaving the agency needs to be replaced. Therefore, the rate of hiring women is equal to the rate of women leaving.
Now, let's introduce some variables to represent the quantities involved:
W = Number of women in the agency.
t = Time in weeks.
Next, let's focus on the number of women leaving the agency. Since the total number of employees leaving per week is constant at 100, and women comprise 25% of the total staff, we can say that the number of women leaving per week is (0.25 * 100) = 25.
So, the rate at which women are leaving is given by dW/dt = -25. (Note: The negative sign indicates a decrease in the number of women in the agency.)
Now, let's consider the rate at which replacements are being hired. Since half of the replacements should be women, the rate at which women are being hired is given by (0.5 * 50) = 25.
Therefore, the rate at which women are being hired is given by dW/dt = +25. (Note: The positive sign indicates an increase in the number of women in the agency.)
To combine both rates, we can write the differential equation as:
dW/dt = Rate of women leaving - Rate of women being hired
dW/dt = -25 - 25
dW/dt = -50
Now, we want to express this differential equation in terms of W and t. To do so, we can introduce another variable, N, which represents the total number of employees in the agency.
N = Total number of employees in the agency.
Since the total number of employees in the agency is constant at 5000, we have N = 5000.
Also, we know that the number of employees leaving per week is constant at 100. Therefore, we can express the total number of employees as N = 5000 - 100t.
Since the number of women in the agency is a fraction of the total number of employees, we can write:
W = (25% * N) = 0.25 * (5000 - 100t) = 1250 - 25t
Now, we can differentiate W with respect to t to express dW/dt in terms of W and t:
dW/dt = d/dt (1250 - 25t) = -25
So, the final differential equation is:
dW/dt = -25
This means that the rate at which women are leaving the agency is constant at -25 per week.
The equation dW/dt = 25 - 2W/(100 - t) that you mentioned in your question seems to have a different formulation or consideration of the problem. It would be helpful to have some context or additional information to understand the reasoning behind that equation.