a skull cleaning factory cleans animal skulls of deer buffal

a skull cleaning factory cleans animal skulls of deer, buffalo and other types of animals using flesh-eating beetles. the factory owner started with only 14 adult beetles. after 45 days, the beetle population grew to 56 adult beetles. how long did it take before the beetle population was 14,000 beetles?

It took about ____days.

(Do not round until the final answer. Then round to the nearest whole number as needed.)

Solution

Let us assume that the beetles grow exponentially and that the population growth model is
A = A₀b^t where A₀ is the initial population and A is the population after t days.

Here we have A₀ = 14, A =56, and t = 45 days so that 56= 14b⁴⁵ or, b⁴⁵ = 4. On taking logarithm of both sides, we have 45log b = log 4 so that b = (1/45)log4 = 0.602059991/45= 0.01337911. Let t be the required number of days. Then
x = 14 * 4^(t / 44) 14000= 14(0.01337911)^t or, (0.01337911)^t = 1000. On taking logarithm of both sides, we have t log(0.01337911) = log 1000 or t = 3/ 0.01337911= 224.23 days, say 224 days.


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