The relationship between the elapsed time \( t \), in years, since Fred started studying the tree, and the number of its branches, \( B_{\text {year }}(t) \), is modeled by the following function: \( B_{\text {year }}(t)=20 \cdot(2.5)^{t} \) Complete the following sentence about the monthly rate of change in the number of branches. Round your answer to two decimal places. Every month, the number of branches increases by a factor of

To find the monthly rate of change in the number of branches, we need to convert the yearly growth factor into a monthly growth factor. The yearly growth factor is \( 2.5 \), which means the number of branches multiplies by \( 2.5 \) each year. To find the monthly growth factor, we can use the formula: \[ \text{Monthly Growth Factor} = (2.5)^{1/12} \] Calculating this gives approximately \( 1.096824 \). Rounding to two decimal places, we find that every month, the number of branches increases by a factor of approximately \( 1.10 \). So, the completed sentence would read: Every month, the number of branches increases by a factor of **1.10**. --- Did you know that the concept of exponential growth isn't just confined to trees? It often appears in population dynamics, where the number of individuals can grow exponentially under ideal conditions! For example, bacteria in a culture can double every few hours under the right environment. Furthermore, when modeling real-life situations involving growth, it's crucial to differentiate between continuous and discrete growth—such as using discrete compounding (monthly) for finances versus continuous rates (like percentage growth). Catching common mistakes in exponent laws can really save you from headaches later!