The count in a bacteria culture was 100 after 15 minutes and 1500 after 30 minutes. Assuming the count grows exponentially,

Question

The count in a bacteria culture was 100 after 15 minutes and 1500 after 30 minutes. Assuming the count grows exponentially,

What was the initial size of the culture? 

Find the doubling period. 

Find the population after 110 minutes.   

When will the population reach 15000. 

Answer 1

P(t) = P₀e^rt; 100 = P₀e^15r and 1500 = P₀e^30r; 100/1500 = e^-15r, r = - ln(1/15)/15 = 0.18053668

100 = P₀e^{15 · 0.18053668}; P₀ = 100/e^2.7080502 = 7

So, P(t) = 7e^0.18052668t

Let P(T) = 14 to get double time T. 14 = 7e^0.18053668T; T = ln2/0.18053668 = 3.84 min

P(110) = 7e^{0.18053668·110} = 2,949,628,152;

15000 = 7e^0.18053668t; t = ln(15000/7)/0.18053668 = 42.484 minmin

Answer 2

1500 = 100e^15k

15 = e^15k

ln 15 = 15k

2.708050201 = 15k

0.18053668 ≈ k

100 = a₀e^{(15)(0.18053668)}

2 = e^{(0.18053668)t}