Exponential growth and decay word problems

Learning Objectives Model real-world scenarios involving growth and decay using the appropriate exponential functions. Solve for any variable (initial amount, final amount, rate, or time) in an exponential model. Utilize natural logarithms to efficiently solve for time in both discrete and continuous exponential models. Calculate the doubling time or half-life of a quantity and use it to determine the rate constant. Differentiate between discrete (e.g., annual compounding) and continuous (e.g., population growth) models and apply the correct formula. Calculate and interpret the instantaneous rate of change of an exponential model at a specific point in time using derivatives. Ever wonder how scientists date ancient fossils or how a small investment can grow into a fortune?...

TermDefinitionExample Exponential GrowthA process where a quantity increases at a rate proportional to its current value. The larger the quantity gets, the faster it grows.A bacterial culture that doubles in size every hour. Exponential DecayA process where a quantity decreases at a rate proportional to its current value. The larger the quantity, the faster it shrinks.The value of a new car depreciating by 18% each year. Growth/Decay Rate (r)The percentage change per unit of time, expressed as a decimal. For growth, r is positive; for decay, r is negative.An investment growth rate of 5% per year corresponds to r = 0.05. A radioactive substance decaying at 2% per year corresponds to r = -0.02. Half-LifeThe specific time it takes for a substance undergoing exponential decay to decrease to h...

General Exponential Model (Discrete Intervals) A(t) = A_0 (1 + r)^t Use this for growth (r > 0) or decay (r < 0) that occurs at fixed intervals (e.g., annually, monthly, quarterly). A(t) is the amount after time t, A_0 is the initial amount, r is the rate per interval, and t is the number of intervals. Continuous Growth/Decay Model A(t) = A_0 e^{kt} Use this for phenomena that grow or decay continuously. A(t) is the amount at time t, A_0 is the initial amount, k is the continuous growth rate constant (k > 0 for growth, k < 0 for decay), and t is time. Rate of Change (Derivative of Continuous Model) \frac{dA}{dt} = A'(t) = k \cdot (A_0 e^{kt}) = k \cdot A(t) This formula gives the instantaneous rate of change at any time t. It shows that the rate of c...