# The count in a bacteria culture was 200 after 15 minutes and 1600 after 30 minutes. Assuming the count grows exponentially,?

What was the initial size of the culture?

Find the doubling period.

Find the population after 115 minutes.

When will the population reach 11000.

Relevance
• Let n=n0 exp(at). Use this after 15 minute it is 200. 200=n0 exp(a 15). Similarly 1600=no exp(a30)

Divide one by the other. 1600/200= exp(a(30-15)),  8= exp (15 a)

ln 8= 15 a, a=0.14. Now use this to find n0. n0= 200*exp(-0.14*15)=24.5. It started with 25 bacterias.

• Login to reply the answers
• Let B(t) = a.bᵗ....where a and b are constants.

When t = 15, B(t) = 200 so,

a.b¹⁵ = 200....(1)

When t = 30, B(t) = 1600 so,

a.b³⁰ = 1600...(2)

(2) ÷ (1) => b¹⁵ = 8

Then, (1) => a(8) = 200

Hence, a = 25....initial population size

Therefore, B(t) = 25(8)ᵗ/¹⁵

Doubling first occurs when B(t) = 50

i.e. 50 = 25(8)ᵗ/¹⁵

=> (8)ᵗ/¹⁵ = 2

so, (t/15)ln8 = ln2

Then, t = 15ln2/ln8 => 5 minutes

Checking gives, 5 mins => 50, 10 mins => 100, 15 mins => 200,...e.t.c.

After 115 minutes we have:

B(115) = 25(8)¹¹⁵/¹⁵ => 209,715,200

For the population to be 11000 we have:

25(8)ᵗ/¹⁵ = 11000

so, (8)ᵗ/¹⁵ = 440

Hence, t = 15ln(440)/ln8 => 44 minutes

:)>

• Login to reply the answers