A culture of bacteria triples every 𝟕𝟕 minutes. How long will it take a culture originally consisting of 𝟐4 bacteria to grow to a population of 𝟏𝟎𝟎 𝟎𝟎𝟎 bacteria? Round to one decimal place, if necessary.
You know that y = 24*2^(x/77)
just solve
24*2^(x/77) = 100000
2^(x/77) = 25000/3
x/77 = log2(25000/3)
Now finish it up.
oops. 100000/24 = 12500/3
Count = a(3)^t, where t is a multiple of 77 minutes
100,000 = 24(3)^t
3^t = 4166.666..
take log of both sides, and follow log rules
t = log 4166.6666../log3 = 7.5867 multiples of 77
= 584.178 minutes
= 9.736 hours or 9 hours and appr 44 minutes
I could have used k/77 hours as my exponent and
found k to be 584.178 minutes, (the actual calculations would remain the same)
nice catch, Reiny. I misread the growth factor. But I'm sure you caught that, right, Sara?
To solve this problem, you need to perform a growth calculation using exponential growth formula. Here's how you can do it step-by-step:
1. Start with the initial population of bacteria, which is 24.
2. Determine the growth rate by observing that the population triples every 77 minutes.
3. Find the number of growth periods needed to reach the target population of 1,000,000 bacteria.
Let's calculate the growth rate:
Divide the time it takes for the population to triple (77 minutes) by the number of tripling periods. In this case, each period is 77 minutes:
Growth rate = 77 minutes / 1 tripling period
The growth rate is the time it takes for the population to triple. In this case, it is 77 minutes.
Next, calculate the number of growth periods needed to reach the target population of 1,000,000 bacteria:
(Number of growth periods) = log(base 3)(target population / initial population)
Plug in the values into the formula:
(Number of growth periods) = log(base 3)(1,000,000 / 24)
Using a calculator to evaluate this expression, you should find that the number of growth periods is approximately 9.3.
Finally, multiply the number of growth periods by the time it takes for the population to triple (77 minutes per period) to find the total time it will take to reach the target population:
Total time = (Number of growth periods) * (time per growth period)
Total time = 9.3 growth periods * 77 minutes per growth period
Round to one decimal place if necessary. This will give you the answer to how long it will take for the bacteria population to reach 1,000,000.