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 15 adult beetles. After 26 days, the beetle population grew to 45 adult beetles. How long did it take before the beetle population was 15,000 beetles? It took about days (Do not round until the final answer. Then to the nearest whole number as needed.)

Solution

The number of flesh eating beetles increased from 15 to 45 ( i.e. tripled ) in 26 days. If we assume tripling every 26 days, then the increasing population is a geometric series with 15 as the 1^st term and the common ratio 3. If 15000 is the nth term then 15(3)^n-1 = 15000 or, 3^n-1 = 1000 = (10)³ Taking logarithms of both the sides, we have (n-1) log 3 = 3 log10 = 3 or, (n-1)* 0.477121254 = 3 so that n -1 = 3/(0. 477121254) = 6.29 ( approx) so that n = 1 + 6.29 = 7.29 . Now, the 2^nd term of the series represents first tripling, the 3^rd term represents 2^nd tripling and so on. Therefore, the 7.29^th term represents 6.29 th tripling which will happen after 26*6.29 = 163.54 days = 164 days ( on rounding off to the nearest whole number) Thus the answer is it took about 164 days.