Problem 12 - Algebra A Combined Function

Question

Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Decay Rate } \\ \text { per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Decay } \end{array} \\ \hline 325,000 & 29 \% & 31 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The final amount after 31 years is 74.

Step 1: Understanding the Formula

We will use the exponential decay formula: \( A = P(1 - r)^t \) where \( A \) is the final amount, \( P \) is the original amount, \( r \) is the decay rate in decimal form, and \( t \) is the time in years. In this problem, \( P = 325,000 \), \( r = 0.29 \), and \( t = 31 \).

Step 2: Convert Decay Rate to Decimal

The decay rate given is 29%. Express this as a decimal by dividing by 100: \( r = 0.29 \).

Step 3: Apply the Exponential Decay Formula

Substitute the known values into the formula: \( A = 325,000 \times (1 - 0.29)^{31} \). Simplifying inside the parenthesis gives \( A = 325,000 \times (0.71)^{31} \).

Step 4: Calculate (0.71)^31

Calculate the power using a calculator: \((0.71)^{31} = 0.0002277\).

Step 5: Calculate the Final Amount

Multiply the original amount by the power calculated: \( A = 325,000 \times 0.0002277 \). This gives \( A = 73.8025 \).

Step 6: Round the Final Amount

The final amount \( A \) is 73.8025. Round this to the nearest whole number to get 74.

Key Concepts

Exponential Functions and Their Role

Exponential functions play a crucial role in various fields, notably in modeling scenarios where change occurs at a constant percentage rate. They can show how quantities increase or decrease exponentially over time. The standard exponential decay formula often takes the form:

• \( A = P(1 - r)^t \) where:
• \( A \) is the final amount,
• \( P \) is the initial amount,
• \( r \) is the rate of decay,
• \( t \) is the time.

By understanding these components, you are prepared to analyze situations involving exponential functions, whether they depict growth or decay. In our exercise, the focus is on decay, using this function to study how quantities decrease over time.

Understanding Percentage Decay

Percentage decay refers to a reduction in quantity that occurs at a constant percentage each year or over each time interval. In the given exercise, the original amount is subject to a 29% decay rate annually.

• To convert the percentage to a decimal, divide by 100: \( r = \frac{29}{100} = 0.29 \).
• This decay indicates that each successive year's amount is only 71% (100% minus 29%) of the previous year's.

Understanding how percentage decay works helps you apply it to real-world problems, whether predicting the depreciation of an asset or modeling population decline. It's essential to convert percentages into decimals when using mathematical formulas.

Mathematical Modeling with Exponential Decay

Mathematical modeling using exponential decay is a powerful tool to predict how quantities decrease over time. Models like the exercise's formula can simulate scenarios in science, economics, and engineering.

• The exercise demonstrates this by using the exponential decay formula, modeling the decrease of an initial amount of 325,000 over 31 years at a 29% decay rate.
• This method helps visualize the exponential nature of decline, making it easier to anticipate future values or understand historical trends.

In practice, these models require careful setup, ensuring that the decay rate is accurately converted and the correct initial values are used. These steps ensure the model remains relevant and reflective of actual phenomena.

Problem-Solving in Algebra with Exponential Decay

Solving problems involving exponential decay in algebra requires a structured approach. As seen in the exercise, we follow specific steps to arrive at the solution:

• Identify the variables: original amount \( P \), decay rate \( r \), and time \( t \).
• Convert any percentage decay rates into a decimal before calculating.
• Plug these values into the exponential decay formula \( A = P(1 - r)^t \).
• Perform calculations step by step, especially the exponential part \((1 - r)^t \).
• Round off calculations suitably, to the nearest whole number if required.

This methodical approach not only ensures accuracy but also enhances your ability to apply these concepts to any similar decay-related problems. With practice, algebraic manipulation of exponential decay formulas becomes intuitive, supporting strong problem-solving skills.