Write a slope field for the differential
equation.
Corporate administrators notice that 13%
of their employees retire each year, while
2,500 are newly hired. What is the long
term effect if they initially have 30,000
employees?
To write a slope field for a differential equation, we first need to determine the equation itself. In this scenario, the differential equation describes the change in the number of employees over time due to retirement and new hires.
Let's represent the number of employees as "N" and time as "t". We know that 13% of the employees retire each year, which means the rate of decrease in the number of employees due to retirement is 0.13N. Additionally, 2,500 new employees are hired each year, resulting in an increase rate of 2,500.
Therefore, the differential equation can be written as:
dN/dt = -0.13N + 2,500
To create a slope field, we can plot several points on a graph representing different values of N and t. At each point, we can find the slope (change in N over change in t) according to the differential equation.
For instance, let's consider the following points:
- N = 30,000, t = 0 (the initial number of employees and time)
- N = 25,000, t = 1 (after the first year)
- N = 23,950, t = 2 (after the second year)
- N = 23,320, t = 3 (after the third year)
- N = 23,073, t = 4 (after the fourth year)
For each point, we find the slope by substituting the values of N and t into the differential equation.
Now, connecting these points with small line segments (arrows) representing the slope at each point will create the slope field. This field helps visualize the direction and magnitude of change in N over time.
As for the long-term effect, we can observe from the slope field that initially, the number of employees will decrease due to retirements exceeding new hires. However, over time, the rate of decrease will slow down and eventually reach a steady state or equilibrium point. This point represents the long-term effect, where the number of employees remains relatively constant.