A bacteria culture has an initial population of 200 bacteria at 9:00 AM and, in the presence of sufficient nutrients, the population doubles every 25 minutes. At approximately what time will there be 100,000 bacteria?
In that case you could proceed like this
after 25 minutes --- 400
after 50 minutes --- 800
after 75 minutes --- 1600
after 100 minutes --- 3200
after 125 minutes --- 6400
after 150 minutes --- 12800
after 175 minutes --- 25600
after 200 minutes --- 51200
after 225 minutes --- 102400 , we needed 100,000, so just short of 225 minutes would do it
my answer using logs would be 224.14 minutes
so 225 minutes would be the closest you could get not using logs,
and 225 minutes after 9:00 am would be 12:45 (I added 3 hrs and 45 minutes)
The course I'm taking does not use log..
To find out at what time there will be 100,000 bacteria, let's break down the problem step by step.
We know that the bacteria population doubles every 25 minutes. So, after 25 minutes, there will be 200 * 2 = 400 bacteria.
Similarly, after another 25 minutes (50 minutes in total), the population will double again to 400 * 2 = 800.
We can observe that the population doubles every 25 minutes. Therefore, to find the number of periods (25-minute intervals) needed to reach 100,000 bacteria, we can set up the following equation:
200 * (2 ^ n) = 100,000
In this equation, 'n' represents the number of periods.
To solve this equation, we can divide both sides by 200:
2 ^ n = 100,000 / 200
2 ^ n = 500
To get rid of the exponential, we can take the logarithm (base 2) of both sides:
log2(2 ^ n) = log2(500)
n = log2(500)
Using a calculator or any logarithm solver, we can find that log2(500) is approximately 8.9658.
Since 'n' represents the number of periods (25-minute intervals), we round up to the next whole number to determine that it will take 9 periods to reach 100,000 bacteria.
Therefore, it will take 9 * 25 minutes = 225 minutes.
Adding the initial time of 9:00 AM, we can calculate the approximate time when there will be 100,000 bacteria:
9:00 AM + 225 minutes = 1:45 PM
So, approximately at 1:45 PM, there will be 100,000 bacteria in the culture.
number = 200(2)^(t/25), where t is in minutes, so ...
200(2^(t/25) ) = 100000
2^(t/25) = 500
take logs of both sides and use your log rules:
t/25 log2 = log 500
t/25 = log500/log2
t = 25(log500/log2)
= .... minutes