A biology student finds a large glass bottle which can be used to grow a bacteri

Question

A biology student finds a large glass bottle which can be used to grow a bacterial culture. She has a bacterial culture that doubles in size every minute and with the amount that she currently has, she calculates that the bottle will be full in one hour. She sterilizes the bottle and places the culture in the bottle at 10 am. Please show your work so I can understand what you did to get the answer. Thank you in advance.

a) At what time will the bottle be half full? why?

b) Let's say that the bottle is "almost empty" when the culture is contained in less than 1% of the volume of the bottle. Approximately what percentage of the time is the bottle "almost empty?

c)After placing the culture in the first bottle, the biologist finds a bottle four times as large (in volume) as the first bottle. If she places the same amount of bacteria in this bottle, at what time will it be full?
d) Think of the world population as a culture and the earth as a bottle. It takes the population of the erath about 35 years to double at current reproduction rates. What do these results say about the world population with respect to growth and "carrying capacity" [you may have to goole a little for the meaning of this term]. What if we find another planet to which we can "migrate"? I am interested in whether you can make a connection between this problem with the bottle and the earth and it's population. A few thoughtful sentence will suffice. [there is no right or wrong anser].
A biology student finds a large glass bottle which can be used to grow a bacterial culture. She has a bacterial culture that doubles in size every minute and with the amount that she currently has, she calculates that the bottle will be full in one hour. She sterilizes the bottle and places the culture in the bottle at 10 am. Please show your work so I can understand what you did to get the answer. Thank you in advance.

a) At what time will the bottle be half full? why?

b) Let's say that the bottle is "almost empty" when the culture is contained in less than 1% of the volume of the bottle. Approximately what percentage of the time is the bottle "almost empty?

c)After placing the culture in the first bottle, the biologist finds a bottle four times as large (in volume) as the first bottle. If she places the same amount of bacteria in this bottle, at what time will it be full?
d) Think of the world population as a culture and the earth as a bottle. It takes the population of the erath about 35 years to double at current reproduction rates. What do these results say about the world population with respect to growth and "carrying capacity" [you may have to goole a little for the meaning of this term]. What if we find another planet to which we can "migrate"? I am interested in whether you can make a connection between this problem with the bottle and the earth and it's population. A few thoughtful sentence will suffice. [there is no right or wrong anser].

Explanation / Answer

Problem A
-----------------

Without doing any math at all, the *question* tells
us that "the bottle will be full in 1 hour". If she
starts the experiment at 10AM, then by the definition
in the question, it will be full at 11AM.



Problem B
----------------
Since the culture doubles every minute, it follows
it increases by

2^t

in "t" minutes.

To go from 1% to 100%, it would increase by a factor
of 100, so solve:

2^t = 100
ln(2^t) = ln(100)
t * ln(2) = ln(100)
t = ln(100)/ln(2)
t = 6.64386 minutes

Therefore, if it took 60 minutes to fill the
flask, it would take 60-6.6 = 53.4 minutes
to fill it up to 1%

Putting this in the form the question asked,
it would spend 53.4/60 = 89.0% of the time
below 1% full.

Then the "last" 99% of the bottle would be filled
in those remaining 6.6 minutes.


Problem C
----------------
Since the new bottle is 4 times as large, it would
take 2 extra minutes to fill it, since:

2^t = 4
t = log_base_2(4)
t = 2

Therefore, it would take a total of 60 + 2 = 2 minutes
to fill. If it was started at 10:00, it would be full
at 11:02.

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