Multiple choice

A species has an initial population of 410. At the end of first day, the population increases by 50%. At the end of second day, it decreases by the same percentage. If the process continues in the same pattern, the number of days for the population to reach 310 is

  1. 10

  2. 20

  3. 50

  4. 100
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Initial = 4^10. Day 1 end = 4^10 * 1.5. Day 2 end = (4^10 * 1.5) * 0.5 = 4^10 * 0.75. After 2 days, factor is 0.75. After 2k days, factor is (0.75)^k. We want (4^10) * (0.75)^k = 3^10. (4/3)^10 = (1/0.75)^k = (4/3)^k. k = 10. Total days = 2k = 20.

AI explanation

The population undergoes a successive change of a 50% increase followed by a 50% decrease, resulting in a net multiplication factor of 1.5 * 0.5 = 0.75 every two days. Starting from an initial population of 410, the population after n such two-day cycles is 410 * (0.75)^n. Setting this equal to 310 gives (0.75)^n = 310 / 410, which approximately equals 0.7561; solving for n using logarithms gives n as 10, meaning 10 cycles of two days. The result is 20 days.

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