Exponential growth and decay: word problems

Learning Objectives Identify whether a word problem describes exponential growth or decay. Translate the conditions of a word problem into an appropriate exponential model (e.g., A(t) = P(1+r)^t, A(t) = Pe^(rt), or half-life/doubling time models). Solve for a final amount given the initial amount, rate, and time. Solve for an initial amount given the final amount, rate, and time. Use logarithms to solve for the time (t) required to reach a certain value. Calculate and interpret the growth or decay rate (r) from given data points. Model and solve problems involving the concepts of half-life and doubling time. Ever wonder how a single social media post goes viral, or how the value of a collector's item skyrockets over time? 📈 That's the power of exponential growth...

TermDefinitionExample Exponential GrowthA pattern where a quantity increases by a constant percentage over equal time intervals. The rate of growth is proportional to the current amount.A bacterial culture that doubles in size every hour. Exponential DecayA pattern where a quantity decreases by a constant percentage over equal time intervals. The rate of decay is proportional to the current amount.A new car losing 15% of its value each year. Growth/Decay Factor (b)The base of the exponential function, which determines the rate of change. For growth, b = 1 + r > 1. For decay, b = 1 - r, where 0 < b < 1.For a 5% annual growth, the growth factor is b = 1 + 0.05 = 1.05. For a 10% annual decay, the decay factor is b = 1 - 0.10 = 0.90. Growth/Decay Rate (r)The percentage of increase or...

General Exponential Growth/Decay Formula A(t) = P(1 + r)^t Use this for situations where growth or decay occurs at discrete intervals (e.g., annually, monthly). 'P' is the initial amount, 'r' is the rate per interval (positive for growth, negative for decay), 't' is the number of intervals, and 'A(t)' is the amount after time t. Continuous Growth/Decay Formula A(t) = Pe^(rt) Use this when growth or decay is happening continuously, not in steps. 'P' is the initial amount, 'e' is Euler's number (approx. 2.718), 'r' is the continuous rate (positive for growth, negative for decay), and 't' is time. Half-Life Formula A(t) = P(1/2)^(t/h) Specifically for radioactive decay or similar proces...