91Ó°ÊÓ

In biology, exponential decay often describes how populations decrease over time. Exponential decay shows how a quantity decreases at a rate proportional to its current value. In this exercise, the formula used to model the animal's weight loss is: \[ W = W_{0} e^{-0.009t} \] Here, \( W \) represents the current weight, \( W_{0} \) is the initial weight, and \( t \) is the time in days. The term \( e^{-0.009t} \) accounts for the exponential decay factor, illustrating how rapidly or slowly the weight is lost each day.

• An important feature of the exponential function is its continuous nature, meaning the weight change occurs every instant.
• The negative sign in the exponent indicates decay — that is, reduction.

This function helps predict future weight based on a current rate of loss. Understanding these small, instantaneous changes helps us to model and predict biological processes better. By using the right decay constant (like \(0.009\) here), we can accurately depict real-world scenarios.

A natural logarithm (ln) is the logarithm to the base of the mathematical constant \( e \), which is approximately equal to 2.71828. In this exercise, we use the natural logarithm to better understand the rate of decay from the formula: \[ \frac{W}{W_{0}} = e^{-0.009t} \] Taking the natural logarithm of both sides simplifies complex exponential relationships: \[ \text{ln}\bigg(\frac{W}{W_{0}}\bigg) = -0.009t \]This helps in translating the exponential decay model to a linear form, making calculations easier.

• The natural logarithm converts exponential problems into more straightforward multiplication, improving our ability to analyze rates and changes.
• Using the natural logarithm, we can confirm or calculate the rate of decay precisely.

For example, plugging \( t = 1 \) day in the equation gives us an immediate way to determine the daily decay constant and understand how exponents influence the decay.

In this problem, we are primarily concerned with both daily and long-term weight loss percentages. The percentage loss helps to quantify the decrease in weight clearly. From the problem, the calculated daily decay rate is approximately 0.9%. Here's how this works: After one day, the weight ratio \( \frac{W}{W_{0}} \) is approximately 0.991, which translates to 99.1% of the original weight remaining. Therefore, the percentage lost is:\[ 100\text\bg - 99.1\text\bg = 0.9\text\bg \] For long-term calculations, such as over 30 days:\[ W = W_{0} e^{-0.009 \times 30} \] Evaluating \( e^{-0.27} \) gives about 0.763. This means that only 76.3% of the weight remains, thus:

• The total weight lost after 30 days is \( 100\text\bg - 76.3\text\bg = 23.7\text\bg \).
• Understanding percentage loss is crucial for tracking biological processes over time, assessing the health and survival chances of organisms, or predicting the outcomes of environmental changes.

This understanding makes it easier to visualize and communicate how exponential decay impacts weight over time.