SOLUTION: The count in a bateria culture was 400 after 15 minutes and 2000 after 35 minutes. Assuming the count grows exponentially, What was the initial size of the culture? Fi

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Question 1108963: The count in a bateria culture was 400 after 15 minutes and 2000 after 35 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 10000.

Found 2 solutions by josgarithmetic, greenestamps:
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
y=a%2Ab%5Ex, x in minutes, y in count of bacteria

system%28400=a%2Ab%5E15%2C2000=ab%5E35%29
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log%28%28400%29%29=log%28%28a%29%29%2B15%2Alog%28%28b%29%29
log%28%282%5E2%29%29%2Blog%28%2810%5E2%29%29=log%28%28a%29%29%2B15log%28%28b%29%29
2%2Alog%28%282%29%29%2B2=log%28%28a%29%29%2B15%2Alog%28%28b%29%29

log%28%282%2A10%5E3%29%29=log%28%28a%29%29%2B35%2Alog%28%28b%29%29
log%28%282%29%29%2B3=log%28%28a%29%29%2B35%2Alog%28%28b%29%29



20%2Alog%28%28b%29%29=3-2%2Blog%28%282%29%29-2%2Alog%28%282%29%29

20%2Alog%28%28b%29%29=1-log%28%282%29%29

log%28%28b%29%29=%281-log%28%282%29%29%29%2F20

log%28%28b%29%29=0.0349485

highlight%28b=1.0838%29
-
y=a%2A%281.0838%29%5Ex
400=a%281.0838%29%5E15
a%2A3.437=400
a=400%2F3.437
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highlight%28a=116%29-----initial population

Equation to Model Example: highlight_green%28y=116%2A%281.0838%29%5Ex%29

Answer by greenestamps(13200) About Me  (Show Source):
You can put this solution on YOUR website!


y+=+ab%5Ex

2000+=+ab%5E35
400+=+ab%5E15

Divide the two equations, eliminating a:

5+=+b%5E20
b+=+5%5E%281%2F20%29+=+1.083798387
Keep at least 4 or 5 decimal places if you want your answers to the later parts of the problem to be accurate.

Question 1: initial population.

The initial population is the population after 15 minutes, divided by b^15:

y+=+400%2Fb%5E15+=+119.6279
This of course is a nonsensical answer; bacteria are counted in whole numbers. However, again you need to keep several decimal places in order for your answers to be accurate.

Answer 1: The initial population was a = 119.6279.

Question 2: doubling time

b is the growth factor each minute; you want to know how many minutes it takes for the growth to be double:
b%5Ex+=+2
x%2Alog%28b%29+=+log%282%29
x+=+log%282%29%2Flog%28b%29+=+8.36153

Answer 2: the doubling time is 8.36153 minutes.

Question 3: the population after 110 minutes

ab%5E110+=+835925

Answer 3: the population after 110 minutes is 835925.

Question 4: the time when the population reaches 10000.

ab%5Ex+=+10000
b%5Ex+=+10000%2Fa
x%2Alog%28b%29+=+log%2810000%2Fa%29+=+4-log%28a%29
x+=+%284-log%28a%29%29%2Flog%28b%29+=+55

Answer 4: the population reaches 10000 at 55 minutes.

Note we could have answered question 4 without doing any difficult calculations. The population increased by a factor of 5, from 400 to 2000, in 20 minutes (between 15 minutes and 35 minutes). 10000 is 5 times 2000, so a population of 10000 will be reached 20 minutes after it was 2000; 20 minutes after 35 minutes is 55 minutes.