The population of a colony of wasps is increasing at the rate of 3.5% each week. If its population in week 1 is 500, then the week in which the number of wasps would first exceed 1000 would be:

Question

BlackTom AI · Accepted Answer

Exponential growth with a weekly multiplier 1.035 applies: population in week n is P(n) = 500 · (1.035)^(n-1) because week 1 corresponds to exponent 0. Step-by-step reasoning and calculation: 1. Set up the inequality for first week the population exceeds 1000: 500 · (1.035)^(n-1) > 1000 (1.035)^(n-1) > 2 2. Solve using logarithms (equality to find threshold): (n − 1) = ln(2) / ln(1.035) Compute the logs (intermediate values shown): ln(2) ≈ 0.693147 ln(1.035) ≈ 0.034392 So (n − 1) ≈ 0.693147 / 0.034392 ≈ 20.16 Therefore n ≈ 21.16. Since n must be an integer week and we need the first week that strictly exceeds 1000, take the next integer: n = 22. 3. Verify by direct computation of nearby weeks: - Week 20 (n = 20): P = 500 · 1.035^(19) ≈ 500 · 1.922 ≈ 961 < 1000 - Week 21 (n = 21): P = 500 · 1.035^(20) ≈ 500 · 1.989 ≈ 994.5 < 1000 - Week 22 (n = 22): P = 500 · 1.035^(21) ≈ 500 · 2.059 ≈ 1029.5 > 1000 Why the correct answer is correct: Because the exponential growth factor reaches a doubling (factor 2) between weeks 21 and 22; the first integer week where population strictly exceeds 1000 is week 22. Why other options are incorrect: - 20th and 21st give populations below 1000 (≈961 and ≈994.5 respectively). - 23rd is after the first exceeding week (it is >1000, but not the first time). Conclusion: The population first exceeds 1000 in the 22nd week; final answer A (22nd) matches the provided answer.

The population of a colony of wasps is increasing at the rate of 3.5% each week. If its population in week 1 is 500, then the week in which the number of wasps would first exceed 1000 would be:单项选择题

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