Question - Population Growth Calculation with Doubling Time
Solution:
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 doublingsTo 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 = 64Thus, 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 hoursNext, we'll calculate how many doublings occur in 504 hours:504 hours / 5 hours per doubling = 100.8 doublingsHowever, 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:2^100This 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.
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