A strain of bacteria multiplies such that the ratio of its population in any two consecutive minutes is constant. If the bacteria grows from a population of 5 million to 40 million over the course of an hour, by what factor does the population increase every 10 minutes?
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This is basically a geometric sequence problem, right? The population multiplies by a constant factor each minute, and over 60 minutes it goes from 5M to 40M. So the total factor over 60 minutes is 8. Then for 10 minutes, it's the 6th root of 8. But that's not any of the answer choices... what am I missing?
Wait, 5 to 40 is times 8, yes. But 60 minutes is 6 intervals of 10 minutes, so factor per 10 min = 8^(1/6). That simplifies to (2^3)^(1/6) = 2^(1/2) = sqrt(2). So answer B. I made the same mistake initially, forgot to simplify.
Ah, got it. I was trying to match the raw root to the choices without simplifying. Thanks!
I got sqrt(2) but I did it by setting up 5 * r^60 = 40, so r^60 = 8, then r^10 = (r^60)^(1/6) = 8^(1/6) = sqrt(2). Is that the standard approach? Seems a bit convoluted. Also, why is choice E sqrt[5]{5}? That's weird.
Yeah, that's fine. E is probably a trap for people who think 60/10 = 5 intervals instead of 6. Or maybe they misread the growth period.
This one took me a while because I kept thinking the growth factor per minute was something I needed to find. But you don't need it. Just recognize that 60 minutes is 6 ten-minute chunks, so the factor over an hour is the 10-minute factor raised to the 6th power. So 10-min factor = 8^(1/6) = sqrt(2). Pretty neat.
I chose D (2) at first because I thought 5 to 40 is times 8, and since 60 minutes is an hour, maybe it doubles every 10 minutes? But that would be 2^6 = 64 times, not 8. So yeah, B is correct. Good question, made me think about exponents.
Same here! I think the trap is to assume it doubles because 5 to 40 is 3 doublings, but that's over the whole hour, not per 10 minutes.
What it tests
Your ability to spot and extend arithmetic and geometric sequences and work with recursive definitions.
Common trap
Off-by-one errors in indexing (a₀ vs a₁) or confusing an arithmetic difference with a geometric ratio.