A population of bacteria doubles every 3 hours. Starting with 500 bacteria, which expression gives the population after t hours?
A500 + 2t
B500 · 2^t
C500 · 2^(t/3)
D500 · (2/3)^t
Every 3 hours, the population doubles — so after t hours it has doubled t/3 times. The base is 2 (the growth factor), and the exponent is t/3. Option A is a linear model (adding instead of multiplying). Option B would double every 1 hour, not every 3. Option D uses 2/3 as a decay factor, which would produce decreasing values.
Question 2 True / False
After two half-lives have passed, a radioactive substance has largely decayed to zero.
TTrue
FFalse
Answer: False
After each half-life, half of the *remaining* substance decays — not half of the original. After one half-life: 50% remains. After two: 25% remains. After three: 12.5%. The quantity follows A(t) = A_0 · (1/2)^(t/h) and approaches zero asymptotically but never reaches it.
Question 3 Short Answer
A savings account earns 6% annual interest compounded annually. Explain the difference between the growth rate and the growth factor in this context, and identify each.
Think about your answer, then reveal below.
Model answer: The growth rate is 6% (or 0.06 as a decimal) — the percentage increase per period. The growth factor is 1.06 — the number you multiply by each year to get the new balance. Growth factor = 1 + growth rate.
Confusing rate and factor is a persistent error. The rate describes how much is added; the factor describes the multiplier. In A(t) = A_0 · (1 + r)^t, the r is the rate (0.06) and (1 + r) = 1.06 is the factor. Using r = 6 instead of 0.06 is a classic mistake that produces wildly wrong answers.