pic: Mathematical Modeling with Exponential and Logarithmic Functions gress: The movement of the progress bar moy be uneven because questons con be worth more or less (Including zero) depending on your answer The population of a town can be modeled using the formula \( P=10,000 e^{0.02 t} \), where \( t \) is the number of years after 2012 and \( P \) is the town's population. Which of the following equations can be used to find the number of years after 2012 that the population will be 15,000 ?

To find the number of years after 2012 that the population will reach 15,000, we start with the equation \( P = 10,000 e^{0.02 t} \) and set \( P \) to 15,000: \[ 15,000 = 10,000 e^{0.02 t} \] Now, divide both sides by 10,000: \[ 1.5 = e^{0.02 t} \] To solve for \( t \), we can take the natural logarithm of both sides: \[ \ln(1.5) = 0.02 t \] Finally, divide by 0.02 to isolate \( t \): \[ t = \frac{\ln(1.5)}{0.02} \] --- Exponential growth is a fascinating phenomenon seen in everything from populations to finance! The classic example of this can be observed in bacterial growth, where populations double at regular intervals under ideal conditions. By understanding the principles behind exponential functions, we not only predict behaviors but can also design strategies for management and sustainability. When working with exponential equations, one common mistake is forgetting to isolate the exponential expression before applying the logarithm. Some calculations can be tricky, and it’s easy to rush through the steps. Always double-check your arithmetic and ensure each step logically follows the last; a small error can lead to a big mix-up in your final answer!