40 new students come to a class each year. At the end of the year, 10% of the students leave the school, 30% stay in the class and repeat the next year, and 10% stay at the end of the second year and repeat the course.
a) Set up the difference equation.
b) Determine the transfer function of the system.
c) How many students will be in the classroom after 5 years?
d) What will be the maximum number of students in the class?
a) To set up the difference equation, let's denote the number of students in the class at the end of year n as N(n). We can express the number of students in the class at the end of year n+1 based on the given conditions:
N(n+1) = N(n) + 0.3N(n) - 0.1N(n) + 40
The first term, N(n), represents the students who stayed from the previous year. The second term, 0.3N(n), represents the students who repeated the course from the previous year. The third term, -0.1N(n), represents the students who left the school at the end of the year. And finally, the constant 40 represents the 40 new students who join the class each year.
b) To determine the transfer function of the system, we need to convert the difference equation into a transfer function format. Let's assume Z-transform notation, where Z represents the discrete time variable.
Taking the Z-transform of both sides of the difference equation, we get:
Z*N(n+1) = Z*N(n) + 0.3Z*N(n) - 0.1Z*N(n) + (40 * Z^0)
Simplifying this equation, we have:
Z*N(n+1) = (1 + 0.3Z - 0.1Z) * Z*N(n) + 40
Dividing both sides by Z*N(n), we obtain:
Z*(N(n+1) / N(n)) = 1.2 - 0.1/Z + (40 / Z*N(n))
Therefore, the transfer function of the system is:
H(Z) = (N(n+1) / N(n)) = (1.2Z - 0.1 + (40 / Z*N(n))) / Z
c) To determine the number of students in the classroom after 5 years, we can recursively apply the difference equation or transfer function.
Let's assume there are initially N(0) students in the class. After 5 years, we want to find N(5). We start with the initial condition:
N(0) = Initial number of students
Then, we can use the difference equation or the transfer function iteratively to find N(5). Since the transfer function has a recursive nature, we can compute each term until we reach N(5).
d) To find the maximum number of students in the class, we need to analyze the difference equation or transfer function in terms of stability. For this particular scenario, since the transfer function has a constant term of 1.2 and no terms involving Z in the numerator, we can conclude that the system is stable.
Therefore, there is no upper limit to the number of students in the class. The maximum number of students will depend on the annual influx of new students and the fraction of students repeating or leaving the school each year.