Question

In Exercises, find exponential models \(y_{1}=C e^{k_{1} t}\) and \(y_{2}=C(2)^{k_{2} t}\) that pass through the points. Compare the values of \(k_{1}\) and \(k_{2} .\) Briefly explain your results. $$ (0,8),\left(20, \frac{1}{2}\right) $$

Short answer

In solving for the parameters of both models, we find that for both models C=8. The rate constant for the first model is \(k_{1}= -0.06931\) whereas for the second model it’s \(k_{2}= -0.1\) . The rate constant is more negative for the second model, implying a faster decay rate.

Step-by-step solution

Formulate the equations based on given points

For each model, plug the given points (0,8) and (20,1/2) into their respective equations: \[y_{1}=C e^{k_{1} t}\] and \[y_{2}=C(2)^{k_{2} t}\] to obtain two equations for each model.

Solve the equations

For the first model, using (0, 8), when t=0, \(y_{1}=C\), therefore C=8. Substitute C=8 into the second equation (from point (20,1/2)) to solve for \(k_{1}\). Similarly, for the second model, substituting (0, 8) gives C=8. Substitute C=8 into the equation from point (20,1/2) to solve for \(k_{2}\).

Comparison

After solving for \(k_{1}\) and \(k_{2}\), the values can be compared to see the effect of the base of the exponential function on the rate constant.

Key concepts

Exponential Growth and Decay

Exponential growth and decay describe processes that increase or decrease at rates proportional to their current value. This concept is fundamental in modeling a wide range of real-world scenarios, such as population growth, radioactive decay, and interest compounding.

For example, an exponential growth model could be represented by the equation \(y = Ce^{kt}\), where \(C\) is the initial amount, \(e\) is the base of the natural logarithm (approximately 2.71828), \(k\) is the growth rate, and \(t\) is time. In contrast, exponential decay models would have a negative growth rate \(k\), indicating a reduction over time.

For the exercise presented, we are asked to find exponential models that reflect either growth or decay through two specific points. This involves identifying the constants \(C\) and \(k\), which provide the initial value and the rate at which the model grows or decays, respectively. Compounded interest scenarios, population studies, or measuring half-lives of radioactive substances are all real-world scenarios where such calculations are crucial.

Solving Exponential Equations

Solving exponential equations is paramount when working with exponential growth and decay models. These equations take the form of \(y = ab^{t}\), where \(a\) is the initial value, \(b\) is the base representing the rate, and \(t\) is time. When the base \(b\) is \(e\), we usually have continuous growth or decay.

In the context of algebra, we typically solve these equations by isolating the variable, often requiring the use of logarithms to deal with the exponent. Here's a simplified approach:

• Establish the equation based on known values.
• Isolate the exponential expression (if possible).
• Take the logarithm of both sides.
• Solve for the unknown variable.

In our exercise, we use the given points to formulate two equations per model and solve for \(k_{1}\) and \(k_{2}\) by isolating these variables. Solving such equations enables us to better understand the behavior of an entity that changes exponentially over time.

Comparing Exponential Functions

Comparing exponential functions helps us understand how different rates of growth or decay affect the behavior of a model over time. These functions may look similar at first glance, but small changes in their formulas can lead to vastly different outcomes.

To compare exponential functions, we look at the base and the exponent's coefficient. If the base is greater than 1, the function represents growth; if it's between 0 and 1, it represents decay. The coefficient, often indicated as the rate \(k\), tells us how rapid the growth or decay is.

Our task involves comparing two models to see how a different base impacts the rate constant. Despite both models having the same initial value, the bases \(e\) and \(2\) will influence the outcomes. Functions with larger bases grow faster – which is a core principle in comparing exponential functions. By analyzing \(k_{1}\) and \(k_{2}\), we can infer the relative rates of growth or decay for each model and apply this knowledge to predict future behavior or to inform decision-making in practical applications.