Problem 149 - Algebra and Trigonometry Real Mathematics, Real People

Question

The numbers \(y\) of commercial banks in the United States from 2007 through 2013 can be modeled by $$y=11,912-2340.1 \ln t, \quad 7 \leq t \leq 13$$ where \(t\) represents the year, with \(t=7\) corresponding to \(2007 .\) In what year were there about 6300 commercial banks? (Source: Federal Deposit Insurance Corp.)

Step-by-Step Solution

Answer

The year would be approximately 2008 as \(t\) is rounded to the nearest whole number. Note that \(t=7\) corresponds to 2007, hence \(t=10.8006\) corresponds to the year 2007 + 10.8006 = 2017.8006 which when rounded to a whole number denotes the year 2018.

Step 1: Substitute the given number of banks into the equation

Substitute \(y = 6300\) into the equation, leading to \(6300=11912 - 2340.1 \ln t\). In this case, the main goal is to isolate \(t\).

Step 2: Solve for \(\ln t\)

First, rearrange to isolate \(\ln t\) on one side of the equation. This can be done by subtracting 11912 from both sides and then dividing by - 2340.1. The equation becomes \(\ln t = \frac{11912 - 6300}{2340.1}\). Calculating the right hand side gives \(\ln t ≈ 2.3793\).

Step 3: Solve for \(t\)

To solve for \(t\), take the exponential of both sides of the equation. Thus, \(t = e^{2.3793}\). Therefore, \(t≈10.8006\).

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are denoted as \( f(x) = a \cdot b^x \), where \(a\) is a constant, and \(b\) is the base. These functions are crucial in modeling growth and decay processes in various fields like finance, biology, and physics.

In the context of our problem, the function involved is a logarithmic model leading to an exponential calculation. By taking the natural logarithm, \(\ln t\), and solving it, we tap into the inverse relationship between exponential functions and logarithms. This allows us to find accurate solutions in scenarios where direct computation might be challenging.

Understanding exponential functions helps you analyze how quantities evolve over time, such as the number of banks in this exercise. Exponential behavior describes how things grow rapidly or decay rapidly depending on the context.

Equation Solving

Solving equations is the act of finding the value(s) of variables that make the given equation true. In algebra, this usually involves isolating the variable by performing a series of operations to balance the equation on both sides.

In the exercise at hand, we are tasked with finding \(t\), the time variable, when the number of commercial banks is approximately 6300. We start by substituting the value into the logarithmic model and solving for \(\ln t\). This involves basic algebraic steps of rearranging and isolating the logarithmic expression.

Taking these steps effectively:

• Substitute known values into the equation
• Rearrange to isolate the target variable
• Use inverse operations (like exponentiation for logarithms) to solve for the variable

This logical flow ensures that even complex equations become manageable. Once the equation is solved for \(\ln t\), applying the exponential function allows us to find the year precisely.

Mathematical Modeling

Mathematical modeling involves creating abstract representations of real-world processes through equations and functions to predict and understand behaviors and outcomes.

The model provided in the exercise, \(y = 11912 - 2340.1 \ln t\), describes the trend in the number of commercial banks over several years. Models like these help in making data-driven decisions, observing patterns, and forecasting future trends.

The construction of such a model involves:

• Identifying key variables and relationships
• Constructing equations that represent these relationships
• Testing the model against actual data to verify its accuracy

In our example, the model helps deduce an estimate of the year when the number of banks equates to a certain value. This predictive ability is invaluable for businesses, economists, and policymakers. Understanding these models empowers you to interpret data with confidence and apply it to practical scenarios.