Problem 6 - College Algebra with Corequisite Support

Question

For the following exercises, identify whether the statement represents an exponential function. Explain. The value of a coin collection has increased by \(3.25 \%\) annually over the last 20 years.

Step-by-Step Solution

Answer

Yes, it's exponential; the value grows by a constant percentage annually.

Step 1: Understand the definition of an exponential function

An exponential function is of the form \(f(x) = a \, b^x\), where \(a\) is the initial value and \(b\) is the base representing the growth factor. The function grows or decays by a constant percentage rate over a constant period.

Step 2: Analyze the coin collection statement

The statement says that the value of a coin collection increases by \(3.25\%\) annually. This implies each year, the value is multiplied by a growth factor influenced by this percentage.

Step 3: Determine if the growth is exponential

To determine the growth factor, convert the percentage to a decimal and add it to 1: \(1 + \frac{3.25}{100} = 1.0325\). Therefore, each year the value is multiplied by \(1.0325\), consistent with an exponential growth model.

Step 4: Conclusion

Since the growth of the coin collection is represented by a constant multiplication over equal time intervals, the statement indeed models an exponential function.

Key Concepts

Growth Factor

When we talk about exponential functions, one key component is the **growth factor**. This is the number by which a quantity is multiplied over time. In the context of our exercise, the growth factor is derived from the percentage increase in the value of the coin collection. To calculate it, we take the given percentage increase and convert it to a decimal. For example, a 3.25% increase translates to 0.0325 in decimal form. We then add 1 to this decimal number to find the growth factor: \[ 1 + \frac{3.25}{100} = 1.0325 \]This means each year the coin collection's value is multiplied by 1.0325. This consistent multiplication is characteristic of exponential growth, as each successive value builds on what came before.

Percentage Increase

Understanding the **percentage increase** is crucial in identifying exponential growth. It indicates how much a quantity grows relative to its previous value. In the example with the coin collection, the percentage increase was 3.25% each year. Here's how you can handle it:

• Convert the percentage to a decimal by dividing by 100. For 3.25%, this is 0.0325.
• Add this decimal to 1 to find the growth factor, which in this case is 1.0325.

This process shows us that the annual growth is consistent year after year, a hallmark of exponential growth. The percentage increase helps us quickly gauge the growth rate and shows us why the pattern isn't linear but rather exponential.

Mathematical Modeling

**Mathematical modeling** is a powerful tool that helps in understanding real-world phenomena by translating them into mathematical language. With this, complex processes such as population growth, radioactive decay, or in our case, the appreciation in the value of a coin collection can be neatly captured.In the scenario of the coin collection, modeling is done using an exponential function:\[f(x) = a \, b^x\]Here,

• \(a\) represents the initial value of the collection.
• \(b\) is the growth factor, which we've calculated as 1.0325.
• \(x\) is the number of years.

This approach allows us to predict the value of the coin collection years down the line. Mathematical models provide clarity and facilitate predictions about future growth based on current data.