Calculating Time Intervals in Exponential Growth and Decay

Miacademy & MiaPrep Learning Channel

This instructional mathematics video guides students through the process of interpreting and solving for time variables within exponential growth and decay models. The instructor, Randy, demonstrates how to handle exponential equations where the exponent is a fraction involving time ($t$). He begins with abstract algebraic examples to establish the procedural logic—setting the fractional exponent equal to 1 to isolate the base growth or decay factor. The video progresses to apply these concepts to real-world scenarios, specifically modeling populations. One example involves calculating the time required for a mouse population to increase by a specific factor, and another determines how long it takes for a bacteria population to be cut in half. The video emphasizes understanding the structure of the equation $A(t) = P \cdot (rate)^{t/k}$ and how to extract the time interval $k$. For educators, this video serves as an excellent resource for Algebra I, Algebra II, or Pre-Calculus units on exponential functions. It bridges the gap between abstract formula manipulation and interpreting word problems. It is particularly useful for teaching students how to identify 'doubling time' or 'half-life' directly from an equation's structure without needing logarithms immediately, provided the question asks for the specific factor given in the base.