A certain illness is spreading at a rate of 10% per hour. How long will it take to spread to 1200 people if 3 people are initially exposed?

Answer 1

Assuming exponential growth:

1200 = 3(1.10)^t

400 = 1.10^t

ln 400 = t ln 1.10

5.9914645 ≈ 0.0931

Can you calculate t and answer?

Answer 2

the rate is given:

dN/dt = kN

where N(t) is the population in units of people over time, and where k = 0.1/hour (given), for 10%/hour.

We solve this differential equation by:

dN/dt = kN

⇒

dN/N = kdt

⇒

∫dN/N = k∫dt

where we integrate dN from 3 to 1200 and t from 0 to t (t is the unknown we are trying to find). Note that this is a definite integral since the start and end points are given and to be found.

⇒ (ln N)(from 3 to 1200) = (kt)(from 0 to t)

⇒

ln(1200)-ln(3) = (0.1)t

⇒

t = [ln(1200/3)]/0.1 = (ln400)/0.1 = 59.915 hours