Explanation:
If the population of bees multiplies 8-fold every 5 weeks, we can calculate how long it would take for the population to triple by finding the number of 8-fold multiplications required.
Let's assume the initial population of bees is P.
To triple the population, we need the final population to be 3P.
Since the population multiplies 8-fold every 5 weeks, the number of 8-fold multiplications required can be determined by dividing the final population by the initial population and taking the logarithm to the base 8.
Number of 8-fold multiplications = log(base 8)(final population / initial population)
Number of 8-fold multiplications = log(base 8)(3P / P)
Number of 8-fold multiplications = log(base 8)(3)
Using logarithm properties, we can rewrite this as:
Number of 8-fold multiplications = log(base 8)(2^1.58496)
Number of 8-fold multiplications = 1.58496 * log(base 8)(2)
Number of 8-fold multiplications ≈ 1.58496 * 0.33333 (approximating log(base 8)(2) as 0.33333)
Number of 8-fold multiplications ≈ 0.52798
Since each 8-fold multiplication occurs every 5 weeks, we can multiply the number of multiplications by 5 to get the total time in weeks:
Total time = Number of 8-fold multiplications * 5
Total time ≈ 0.52798 * 5
Total time ≈ 2.64 weeks
Therefore, it would take approximately 2.64 weeks for the population of bees to triple if they multiply 8-fold every 5 weeks.