Answer:
Inicial size of the culture = 69.2246
Doubling period = 7.0902 minutes
Population after 105 minutes = 1,987,397.71
Time for population reaches 12000: 52.7334 minutes
Step-by-step explanation:
First we need to find the exponencial function using the two information given. The model for an exponencial function is:
P = Po * (1+r)^t
Where P is the final value, Po is the inicial value, r is the rate and t is the time. So we have that:
300 = Po * (1+r)^15
1300 = Po * (1+r)^30
Isolating Po in both equations, we have that:
300/(1+r)^15 = 1300/(1+r)^30
(1+r)^30/(1+r)^15 = 1300/300
(1+r)^15 = 4.3333
1+r = 1.1027
r = 0.1027
From the first equation, we can use r to find Po:
300 = Po * (1+0.1027)^15
Po = 300 / (1.1027)^15 = 69.2246
To find the doubling period, we have that P/Po = 2, so:
(1+0.1027)^t = 2
log(1.1027^t) = log(2)
t*log(1.1027) = log(2)
t = log(2)/log(1.1027) = 7.0902 minutes
The population after 105 minutes is:
P = 69.2246 * (1+0.1027)^105 = 1,987,397.71
When the population reaches 12000:
12000 = 69.2246 * (1+0.1027)^t
(1.1027)^t = 12000/69.2246
log(1.1027^t) = log(173.3488)
t*log(1.1027) = log(173.3488)
t = log(173.3488)/log(1.1027) = 52.7334 minutes