A population of bears increased by \( 50 \% \) in 4 years. If the situation is modeled by an annual growth continuously, which formula could be used to find the annual rate according to the exponential gro answer in terms of \( \ln \). Note: When entering natural log in your answer, enter lowercase LN as "In". There is no "natural log' keyboard.

To find the annual growth rate \( r \) for continuous growth modeled by the formula \( P(t) = P_0 e^{rt} \), we first note that a 50% increase means \( P(4) = 1.5P_0 \). Setting up the equation gives us \( 1.5P_0 = P_0 e^{4r} \). By dividing both sides by \( P_0 \) and simplifying, we can take the natural logarithm: \[ \ln(1.5) = 4r \] From this, we can solve for \( r \): \[ r = \frac{\ln(1.5)}{4} \] --- Did you know that continuous growth models aren't only used for populations, but also for things like investments? Just like your bear population, your money can grow continuously! The formula is adaptable; just ensure your rates are clear, and soon you'll see how your initial amount can multiply over time (compounding interest!). Moreover, remember that misunderstanding initial values can lead to miscalculations! It’s vital to set your \( P_0 \) correctly. Be careful not to confuse the increase percentage with the rate itself, as \( r \) represents the actual annual growth rate, not the overall percentage increase in the population over the 4-year span.