Exponential Growth and Decay

Pre-Calculus · Axiom Academy

LESSON Exponential Growth and Decay One model — y = a(1+r)^t — that climbs when a quantity grows by a fixed percent and falls when it shrinks by one, plus the doubling time and half-life hiding inside it. 1. Growth: a Population That Keeps Doubling A town of 10,000 people grows 3% per year . Each year multiplies the population by 1.03 , so after t years y = 10000(1.03)^t . The increase isn't a flat number of people — it's a flat percentage , so the bigger the town gets, the faster it adds people. Watch the curve climb, then accelerate. Doubling time — when does y reach 2a ? 2. Decay: a Sample That Keeps Halving A 160 mg sample of iodine-131 has a half-life of 8 days : every 8 days, exactly half of it is gone. That's the half-life form . The curve falls steeply at first, then flattens — it keeps halving toward the y=0 axis but never reaches it . Watch each 8-day step cut the amount in half. Decay model — base , halves every h As a percent rate: lose per day but always — zero is an asymptote. Because the percentage lost per step is fixed, the time to halve is constant — the mirror image of doubling time. It's independent of how much you start with: 160 mg and 16 mg both lose half their mass in the same 8 days. 3. One Number Decides: the Base b

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