A single-cell amoeba doubles every 4 days. How long would it take one amoeba to produce a population of about 10,000 amoebae?

Question

A single-cell amoeba doubles every 4 days. How long would it take one amoeba to produce a population of about 10,000 amoebae?

Answer 1

So the number of amoebae doubles every cycle (a cycle being four days).  So we just multiply it by 2 every cycle

 

2 x 2 x 2...

 

A simple way to write this is using an exponent with a base of 2

 

2ⁿ

 

where n represents the number of cycles.

 

2¹ = 2 amoebae after one cycle

2² = 4 amoebae after two cycles

2³ = 8 amoebae after three cycles, and so on

 

So you can substitute different numbers for n until you find the correct number of cycles, or you can take the base 2 logarithm of the equation.

 

2ⁿ = 10000

log₂(2ⁿ) = log₂(10000)

n = log₂(10000)

 

Remember, n represents the number of cycles, so once you find it you will have to multiply it by the length of each cycle.

Answer 2

N(t) = 2^t gives number of amoebas after t 4-day periods..Solve N(t) > 10,000.

10,000 < 2^t

taking common logs, 4 < t (log 2)

t > 4/0.30301 

t>13.2877124 (# 4-day periods)

 multiplying by 4 to get # days, t > 53.151 days.