Answer:
x(t) = 2000 - e^(-k*t)
Step-by-step explanation:
Interpretation:
No . infected student = x
Total student = 2000
rate of infected students = dx / dt
not-infected student = 200 - x
The general rate at which student are infected can be expressed as below:
dx / dt = k * ( 2000 - x )
To develop an expression of x(t) we integrate the above expression by separating variables:
dx / (2000 - x ) = k * dt
Now integrate:
[tex]\int\limits{\frac{1}{(2000-x)} } \, dx = k * \int\limits{dt}\\\\- ln (2000-x) = k*t + C\\\\[/tex]
@ t = 0 , infected students x = 0
Hence,
C = - ln (2000)
[tex]- ln (2000-x) = k*t + - ln (2000)\\\\ln (\frac{2000-x}{2000}) = -k*t\\\\ 2000 - x = e^(-kt)\\\\x(t) = 2000 - e ^ (-k*t)[/tex]
