Population Growth Calculation with Doubling Time
The question in the image pertains to exponential growth, more specifically the doubling time of a population.
The doubling time of a population is the time required for that population to double in size at its current growth rate.
The image states:
"The doubling time of a population of flies is 5 hours. By what factor does the population increase in 30 hours? By what factor does the population increase in 3 weeks?"
Let's solve this in two parts:
1. Calculation of the population increase factor in 30 hours:
Given that the population doubles every 5 hours, we can find out how many doubling periods there are in 30 hours by dividing 30 by the doubling time of 5 hours:
30 hours / 5 hours per doubling = 6 doublings
To determine the factor by which the population has increased after 6 doublings, we calculate 2 to the power of 6 (since each doubling multiplies the population by 2):
2^6 = 64
Thus, the population increases by a factor of 64 in 30 hours.
2. Calculation of the population increase factor in 3 weeks:
First, let's convert 3 weeks into hours. There are 24 hours in a day and 7 days in a week, so:
3 weeks * 7 days/week * 24 hours/day = 504 hours
Next, we'll calculate how many doublings occur in 504 hours:
504 hours / 5 hours per doubling = 100.8 doublings
However, part of a doubling period does not result in a full doubling, so we must consider only the complete doublings. Therefore, we will consider 100 doublings.
Now let's calculate the factor for 100 doublings:
2^100
This number is extremely large (roughly a 30-digit number), so it is impractical to write out in its entirety. For simplicity, it suffices to say that the population increases by a factor of 2^100 after 3 weeks.
These calculations assume that the growth can continue without any limitations, which in real-world scenarios may not be possible due to factors like resources and space constraints.
