Finding 'r' In Exponential Decay: A Step-by-Step Guide

Understanding exponential functions is crucial in various fields, from mathematics and physics to finance and biology. One common application is modeling the decay of a substance over time. In this article, we'll delve into how to determine the constant 'r' in an exponential function of the form f(t) = 2e^(r-t), where f(t) represents the amount of a substance at time t. Let's break down the process step-by-step to make it clear and straightforward.

Understanding the Exponential Function

Before we jump into finding 'r', let's first ensure we have a solid grasp of what the function f(t) = 2e^(r-t) represents. This equation models exponential decay, where the amount of a substance decreases over time.

• f(t): This represents the amount of the substance remaining at time t. It's what we're trying to predict or understand based on the model.
• 2: This is the initial amount of the substance at time t = 0. It's like the starting point of our decay process. We know that at the beginning, we had 2 grams of the substance.
• e: This is the base of the natural logarithm, approximately equal to 2.71828. It's a fundamental constant in mathematics and appears frequently in exponential functions.
• r: This is the constant we're trying to find. It influences the rate of decay. A larger negative value of r means a faster decay, while a smaller negative value indicates a slower decay. If r is positive, this models exponential growth instead of decay.
• t: This represents time, usually measured in days in this context. It's the independent variable in our function, and we plug in different values of t to see how the amount of the substance changes over time.

The expression (r - t) in the exponent is crucial. It dictates how the exponential function behaves. Because 't' is increasing over time, a negative 'r' ensures that the overall exponent becomes more negative as time goes on, leading to a decrease in f(t). The term e^(r-t) is the core of the exponential decay, showing how the substance diminishes over time.

In essence, this function describes a scenario where a substance starts at an initial amount of 2 grams and gradually decreases as time passes. The constant 'r' is the key to unlocking the specific rate of this decay, and our goal is to determine its value based on given data points. Understanding each component of the function allows us to approach the problem methodically and make sense of the solution we find. We are essentially reverse-engineering the decay process by using the information we have about the substance's amount at certain points in time.

Step-by-Step Guide to Finding 'r'

Now, let's dive into the process of finding the constant 'r'. We'll use a step-by-step approach to make it easy to follow.

Step 1: Gather the Data

The first step is to gather the data provided. The problem states that the table gives values for the amount of the substance on certain days. This means we'll have pairs of values (t, f(t)), where t is the number of days and f(t) is the amount of the substance in grams. For example, you might have data points like (1, 1.5), (2, 1.1), and so on. The more data points you have, the more accurately you can determine 'r'.

Having multiple data points is particularly useful because it allows you to check the consistency of your 'r' value. If the substance truly follows an exponential decay model, the 'r' value you calculate from different pairs of data points should be relatively consistent. Significant variations in 'r' might suggest that the exponential model isn't a perfect fit for the data, or there might be experimental errors in the measurements.

Consider the practical aspect of data gathering. In a real-world scenario, this data could come from laboratory measurements, where the amount of a radioactive substance is tracked over time, or from observations of population decline in ecological studies. Accurate data collection is paramount because the reliability of your 'r' value directly depends on the quality of the data. Any errors in the measurements will propagate through the calculation and affect the accuracy of the model.

Step 2: Select a Data Point

Next, select one data point (t, f(t)) from the data you've gathered. This means you'll choose a specific day (t) and its corresponding amount of substance (f(t)). For instance, if you have the data point (3, 0.8), you know that on day 3, there were 0.8 grams of the substance remaining. This single data point will be crucial in solving for 'r' because it provides a specific condition that the exponential function must satisfy.

The choice of data point can sometimes impact the ease of calculation or the accuracy of the result. If you have very precise measurements, any data point should yield a similar 'r' value. However, in practical situations, measurements may have some degree of error. In such cases, choosing a data point that is neither too early nor too late in the decay process might be beneficial. Early data points might be closer to the initial condition and thus less affected by the decay process, while later data points might have smaller amounts and be more susceptible to measurement errors.

Consider the implications of your choice. By selecting a particular (t, f(t)) pair, you are essentially anchoring your exponential curve to pass through that point. The 'r' value you calculate will be the one that makes the curve fit this specific condition. Therefore, it's essential to ensure that the chosen data point is reliable and representative of the overall decay process. If there are reasons to suspect some data points (e.g., outliers or measurements taken during a system disturbance), it might be prudent to avoid using them for the initial calculation and save them for later validation.

Step 3: Substitute the Values

Now, substitute the values of t and f(t) from your chosen data point into the exponential function f(t) = 2e^(r-t). This means you'll replace f(t) with the amount of the substance at the chosen time, and you'll replace t with the number of days. For example, if you selected the data point (3, 0.8), you would substitute f(t) with 0.8 and t with 3, resulting in the equation 0.8 = 2e^(r-3). This substitution is a crucial step because it transforms the general exponential function into a specific equation that you can solve for the unknown variable 'r'.

The act of substitution is a fundamental technique in algebra and mathematical modeling. It allows you to tailor a general formula to a specific situation by incorporating known values. In this context, substituting the data point into the equation effectively turns 'r' into the only unknown variable, setting the stage for the next step where we isolate and solve for it.

It is important to be meticulous during the substitution process to avoid errors. Ensure that you are correctly matching the f(t) value with the corresponding time t. A mistake here can lead to an incorrect 'r' value and, consequently, an inaccurate model of the substance's decay. Double-checking your substitution can save time and prevent confusion later on. Also, pay attention to units; ensure that the units of time are consistent (e.g., days) and that the amount of the substance is in the correct units (e.g., grams) to maintain the integrity of the equation.

Step 4: Isolate the Exponential Term

The next step is to isolate the exponential term. In the equation 0.8 = 2e^(r-3), the exponential term is e^(r-3). To isolate it, you'll need to divide both sides of the equation by 2. This gives you 0.8 / 2 = e^(r-3), which simplifies to 0.4 = e^(r-3). Isolating the exponential term is a critical step because it allows us to apply the inverse operation (the natural logarithm) to solve for the exponent, which contains our unknown constant 'r'.

Isolating a term in an equation is a common algebraic technique used to simplify the equation and bring us closer to the solution. By isolating the exponential term, we're essentially preparing the equation for the application of logarithms, which will