SOLUTION: The count in a bacteria culture was 600 after 20 minutes and 1600 after 30 minutes. Assuming the count grows exponentially, What was the initial size of the culture? Fin

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Question 1198940: The count in a bacteria culture was 600 after 20 minutes and 1600 after 30 minutes. Assuming the count grows exponentially,
What was the initial size of the culture?

Find the doubling period.

Find the population after 60 minutes.

When will the population reach 13000.
Answer by MathLover1(20850) About Me  (Show Source):
You can put this solution on YOUR website!

general exponential models y+=+a%2Ab%5Et, where t is the time from the starting moment and b is some arbitrary base

if the count in a bacteria culture was 600 after+20 minutes,we have
600+=+a%2Ab%5E20............eq.1........solve for a
a=600%2Fb%5E20+..........eq.1a

if the count in a bacteria culture was 1600 after+30 minutes,we have
1600+=+a%2Ab%5E30............eq.2, sustitute a

1600+=+%28600%2Fb%5E20+%29%2Ab%5E30............simplify
1600+=+600%2Ab%5E10
1600%2F600=b%5E10
8%2F3=b%5E10
b=root%2810%2C8%2F3%29
then

a=600%2Fb%5E20+..........eq.1a
if 8%2F3=b%5E10=>%288%2F3%29%5E2=b%5E20=>b%5E20=64%2F9
a=600%2F%2864%2F9%29+
a=675%2F8
general exponential models y+=+%28675%2F8%29%2A%28root%2810%2C8%2F3%29%29%5Et
What was the initial size of the culture?
the initial population "a" was675%2F8

The doubling period equation is +b%5Et+=+2
root%2810%2C8%2F3%29%5Et+=+2
log%28root%2810%2C8%2F3%29%5Et%29+=+log%282%29
t%2Alog%28root%2810%2C8%2F3%29%29+=+log%282%29
t=+log%282%29%2Flog%28root%2810%2C8%2F3%29%29+
t=+log%282%29%2Flog%28root%2810%2C8%2F3%29%29+
t=7.06695min


Find the population after 60 minutes.
y+=+%28675%2F8%29%2A%28root%2810%2C8%2F3%29%29%5E60
y+=+819200%2F27
y=30340.74

When will the population reach 13000
13000=+%28675%2F8%29%2A%28root%2810%2C8%2F3%29%29%5Et
t+=+%288+log%284160%2F27%29%29%2Flog%281000%29
t=5.83min