The initial size of a bacteria culture that grows exponentially was 10,000. After 1 day, there are 15,000 bacteria...

Question

 How long will it take for the culture to double in size?

Answer 1

P(t) = P₀·b^t

• P(t) = bacteria population at time t
• P₀ = starting bacteria population = 10,000
• b = growth rate
• t = days

 

P(t) = P₀·b^t

 

At t = 1 day, P(t=1) = 15,000:

 

15,000 = 10,000·b¹

15,000/10,000 = b

1.5 = b

 

P(t) = 10,000·1.5^t

 

Population doubles when P(t) = 20,000:

 

20,000 = 10,000·1.5^t

2 = 1.5^t

log(2) = log(1.5^t)

log(2) = t·log(1.5)    [Log property: log(a^b) = b·log(a)]

log(2)/log(1.5) = t

 

Use you calculator to get the answer.  You can use any base log.

 

Answer 2

T = doubling time in days

 

A(t) = A₀ (2)^t/T,

 

t in days.

 

15,000 = 10,000 (2)^1/T

 

1.5 = 2^1/T

 

log1.5 = (1/T) log 2

 

T = log2 / log1.5 ≅ 1.7095 days