Bacteria Growth Exponentially

Question

A population of bacteria is growing exponentially. If at 1pm there are 600 bacteria and at 3pm there are 720 bacteria, what is the hourly growth rate? How long will it take for the population to triple?

Answer 1

Exponential Growth Equation:

y=y₀e^kt, k > 0

(a) What is the hourly growth rate?

y(t) = y₀e^kt

720 = 600e^2k

720/600 = e^2k

1.2 = e^2k

ln(1.2) = ln(e^2k)

= 2k · ln(e)

= 2k · 1

k = ln(1.2)/2

k ≈ 0.09116...

(b) How long will it take for the population to triple?

y(t) = y₀e^kt

1800 = 600 · e^(ln(1.2)/2)t

1800/600 = e^(ln(1.2)/2)t

3 = e^(ln(1.2)/2)t

ln(3) = ln(e^(ln(1.2)/2)t)

= ln(1.2)/2 · t · ln(e)

= ln(1.2)/2 · t · 1

t = ln(3)/(ln(1.2)/2)

t = (2 · ln(3))/ln(1.2)

t ≈ 12.05137...