The count in a bacteria culture was 300 after 15 minutes and 1300 after 30 minutes. Assuming the count grows exponentially,

What was the initial size of the culture?

Find the doubling period?

Find the population after 105 minutes?

When will the population reach 12000

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