Problem 64 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 63-64: An economist suspects that the following data points lie on the graph of \(y=c 2^{k x}\), where \(c\) and \(k\) are constants. If the data points have three-decimal-place accuracy, is this suspicion correct? $$ \begin{aligned} &(0,-0.3),(0.5,-0.345),(1,-0.397),(1.5,-0.551) \\ &(2,-0.727) \end{aligned} $$

Step-by-Step Solution

Answer

The model \(y=c \cdot 2^{kx}\) does not fit the given data points.

Step 1: Understanding the Exponential Model

The economist suspects that the data follows the formula \(y = c \cdot 2^{kx}\), where \(c\) and \(k\) are constants. We aim to determine values for \(c\) and \(k\) that fit the data and check if all data points satisfy this model.

Step 2: Finding Constants Using Point (0, -0.3)

Substitute the first data point \((0, -0.3)\) into the equation. Since \(x = 0\), the equation becomes \(y = c \cdot 2^{k \cdot 0} = c \cdot 1 = c\). Therefore, \(c = -0.3\).

Step 3: Calculating Value of k

Using the second data point \((0.5, -0.345)\), substitute into \(y = -0.3 \cdot 2^{k \cdot 0.5}\). Solving gives:\[-0.345 = -0.3 \cdot 2^{0.5k}\]\[2^{0.5k} = \frac{-0.345}{-0.3} = 1.15\].Take logarithms:\[0.5k \log 2 = \log 1.15\]\[k \approx 2 \cdot \frac{\log 1.15}{\log 2}\].

Step 4: Verification

Calculate \(k\) from previous step using logarithm values. Approximate:\(\log 2 \approx 0.3010\) and \(\log 1.15 \approx 0.0607\).\[k \approx 2 \cdot \frac{0.0607}{0.3010} \approx 0.403\].Substitute \(k\) and \(c\) back to check other data points.

Step 5: Check Agreement with Other Data Points

For the point \((1, -0.397)\), substitute back:\[-0.397 = -0.3 \cdot 2^{0.403 \cdot 1}\].Similarly, check with \(x = 1.5, 2\). If calculated values of \(y\) closely match actual values, then the model fits.However, checking for \(x = 1.5\) and \(x = 2\) shows significant deviation which indicates model mismatch.

Key Concepts

Exponential Growth

Exponential growth is a process where a quantity increases according to an exponential function. In mathematics, this is often described by the equation \( y = c \cdot 2^{kx} \), where \( c \) is an initial value and \( k \) is the growth rate constant.

• The exponential function implies rapid increase as the value of \( x \) increases, assuming \( k > 0 \).
• Exponential growth is common in natural systems such as populations, nuclear reactions, and financial investments.

In the context of this exercise, the economist hypothesizes that the data points provided exhibit such growth. This means they suspect the data follows the pattern where the rates of increase, relative to the current size, are constant. They aim to determine values of \( c \) and \( k \) to confirm if these points fit the exponential growth model.Understanding this growth type is crucial since it indicates whether an assumption of continual growth or decline over time (e.g., in economics) is valid.

Data Fitting

Data fitting is the process of finding a mathematical function that best describes a set of data points. In this exercise, the aim is to check if the data fits the model \( y = c \cdot 2^{kx} \) by adjusting the constants \( c \) and \( k \).

• By using known data points, we modify \( c \) and \( k \) to minimize the difference between observed values and model predictions.
• In practice, this involves inserting data points into the equation and solving for unknowns. Calculating the residuals, or the differences between observed and predicted values, helps assess fit quality.

In this process, challenges can arise if the model does not sufficiently capture the data's underlying dynamics. For our data, even small deviations in calculations can indicate that the model may not perfectly align. It's important to confirm that mathematical approximations remain accurate enough to satisfy the initial hypothesis.

Mathematical Modeling

Mathematical modeling involves creating equations or simulations to represent real-world scenarios. This enables analysts to predict and understand data patterns and behaviors over time.

• Models provide a simplified representation of complex systems to identify general trends or explain phenomena.
• An effective model, like the exponential model here, should accurately reflect the conditions of the data it's describing.

In the context of this exercise, the economist uses an exponential model to assess economic data points. A model's success lies in its ability to reproduce observed data when values for \( c \) and \( k \) are applied. If a mismatch occurs, as seen when additional points significantly deviate from expected outcomes, this could suggest a need for a different model or further refinement. The choice of the correct mathematical model is crucial for accurate predictions and useful insights.

Constant Determination

Determining constants \( c \) and \( k \) is a vital step in formulating an exponential growth model. These constants define the rate and scale of the growth process described by the function.

• To find \( c \), use a point where \( x = 0 \), leading directly to \( y = c \).
• To determine \( k \), use additional data points by substituting into the model and solving for \( k \), often involving logarithms.

In this exercise, the economist starts with the point \((0, -0.3)\) to find \( c = -0.3 \). Then, using \((0.5, -0.345)\), they calculate \( k \). Logarithmic transformations simplify solving exponential equations when \( k \) appears in the exponent. Once \( c \) and \( k \) are known, you check all points to see if deviations occur, as they can indicate whether these constants satisfactorily describe the entire dataset. This process is essential for ensuring that conclusions drawn from the model are valid and backed by data.