The count in a bateria culture was 500 after 15 minutes and 1600 after 40 minutes. Assuming the count grows exponentially,
What was the initial size of the culture?
Find the doubling period.
Find the population after 65 minutes.
When will the population reach 10000.

Question

The count in a bateria culture was 500 after 15 minutes and 1600 after 40 minutes. Assuming the count grows exponentially,
What was the initial size of the culture?
Find the doubling period.
Find the population after 65 minutes.
When will the population reach 10000.

aum · Answer

$$
N = a * b^t \
t = 15, N = 500 \
500 = a * b^{15} ~~\text{ ------ (1) }\
t = 40, N = 1600 \
1600 = a * b^{40} ~~\text{ ------ (2) }\
\text{(2) / (1):} \
\frac{1600}{500} = b^{40-15} = b^{25} \
b^{25} = 3.2 \
25\ln(b) = \ln(3.2) \
\ln(b) = \frac{\ln(3.2)}{25} = 0.0465 \
b = e^{0.0465} = 1.0476 \
500 = a * (1.0476)^{15} ~~\text{ ------ (1) }\
a = \frac{500}{(1.0476)^{15} } = 248.8 \approx 249  \
N = 249 * (1.0476)^{t} 
$$