Question

Out of a total of N students at a school, the number of students who have seen a new television program increases at a rate proportional to the product of the number of students who have seen the program and the number of students who have not seen the program. If S denotes the number of students who have seen the program at time tt, which of the following differential equations could be used to model this situation, where k is a positive constant

Answer 1

[TeX] (C) \frac{dS}{dt}=kS(N-S) [/TeX]

Explanation:

The given options are:

[TeX] (A)\frac{dS}{dt}=kS \\ (B) \frac{dS}{dt}=kt(N-t)\\ (C)\frac{dS}{dt}=kS(N-S)\\ (D)\frac{dS}{dt}=kS(S-N) [/TeX]

Let the total Number of students be N.

Let the number who have seen the program = S

Therefore, the number of those who have not seen the program = N-S

Product of the number of students who have seen the program and the number of students who have not seen the program=S(N-S)

Since the number of students who have seen the new television program increases at a rate proportional to the product of the number of students who have seen the program and the number of students who have not seen the program.

Therefore:

[TeX] \frac{dS}{dt}=kS(N-S) [/TeX] (where k is a positive constant) is the differential equations which could be used to model this situation.