A hostel has 100 students. On a certain day (consider it day zero) it was found that two students are infected with some virus. Assume that the rate at which the virus spreads is directly proportional to the product of the number of infected students and the number of non-infected students. If the number of infected students on 4ᵗʰ day is 30, then number of infected students on 8ᵗʰ day will be __________.

Solution

Maths sir · 5 steps

Separate the variables, and the ratio infected to non-infected grows by the same factor every 4 days.

1. Set up the equation. Let x be the number infected on day t, so 100 - x are not infected. The spread rate is dx/dt = kx(100 - x), and this separates into dx/x(100-x) = k dt.
2. Integrate. Split with partial fractions: 1/x(100-x) = 1/100(1/x + 1/(100-x)). Integrating both sides and taking exponentials gives the result below, where C is a constant.

   x/(100 - x) = C e¹⁰⁰ᵏᵗ

3. Use the data. At t = 0, x = 2, so C = 2/98 = 1/49. At t = 4, x = 30, so 30/70 = 1/49e⁴⁰⁰ᵏ, which gives e⁴⁰⁰ᵏ = 3/7 × 49 = 21.
4. Day 8. e⁸⁰⁰ᵏ = (e⁴⁰⁰ᵏ)² = 441, so x/(100 - x) = 441/49 = 9. Solving, x = 900 - 9x, so 10x = 900.
5. So. x = 90. On the 8th day, 90 students are infected.

The idea to keep. For logistic growth dx/dt = kx(N - x), the ratio x/(N - x) grows exponentially. Equal time gaps multiply this ratio by the same factor, so work with the ratio, not with x.

Answer from the NTA final answer key. Solution by PrepWiser.