Problem 10 - Algebra A Combined Function

Question

Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Decay Rate } \\ \text { per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Decay } \end{array} \\ \hline 15,000 & 16 \% & 11 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The final amount after 11 years is approximately 2193.

Step 1: Understanding the Exponential Decay Formula

The exponential decay formula is given by \( A = P(1 - r)^t \), where \( A \) is the final amount, \( P \) is the original amount, \( r \) is the decay rate (expressed as a decimal), and \( t \) is the time in years. In this exercise, \( P = 15000 \), \( r = 0.16 \), and \( t = 11 \).

Step 2: Convert the Percent Decay Rate to Decimal

To convert the decay rate from a percentage to a decimal, divide by 100. Therefore, the decay rate as a decimal is \( r = 16\% = 0.16 \).

Step 3: Insert Values into the Exponential Decay Formula

Plug the given values into the decay formula: \( A = 15000 \times (1 - 0.16)^{11} \). Simplify inside the parentheses first.

Step 4: Simplify Inside the Parentheses

Calculate \( 1 - 0.16 = 0.84 \). So, the expression is now \( A = 15000 \times 0.84^{11} \).

Step 5: Calculate \( 0.84^{11} \)

Use a calculator to find \( 0.84^{11} \). This calculation gives approximately \( 0.146197 \).

Step 6: Find the Final Amount

Multiply the result from step 5 by the original amount: \( A = 15000 \times 0.146197 \approx 2193 \).

Step 7: Round to the Nearest Whole Number

The calculated final amount is approximately 2193. Round this to the nearest whole, which remains 2193.

Key Concepts

Exponential Functions

Exponential functions are mathematical expressions where a constant base is raised to a variable exponent. They are ubiquitous in scenarios involving growth or decay, such as population dynamics, radioactive decay, and interest calculations. The general form of an exponential function is \( y = a \, b^x \), where:

• \( y \) is the value of the function at \( x \)
• \( a \) represents the initial value, or starting amount
• \( b \) is the base of the exponential, determining the nature of growth or decay
• \( x \) is the exponent, representing the variable (like time)

For exponential decay, the base \( b \) is a fraction less than 1. This makes the quantity decrease over time, demonstrating a core concept in decay functions.

Decay Rate

The decay rate is crucial to understanding how quickly an amount decreases over time in exponential decay. It represents the fraction of the quantity that is lost per unit of time, expressed commonly as a percent. To use this percent effectively in equations, convert it to a decimal by dividing by 100. This step is essential:

• Given as a percentage for intuitive understanding (e.g., 16%)
• Converted to a decimal for algebraic use (e.g., 0.16)

The decay rate determines the base of the exponent in the function. For instance, if the decay rate is 16%, the base becomes \( 1 - 0.16 = 0.84 \), implying that 84% of the previous amount remains after each time period.

Mathematical Formula

The mathematical formula for exponential decay is given by \( A = P(1 - r)^t \), where:

• \( A \) is the final amount remaining after time \( t \)
• \( P \) denotes the initial amount
• \( r \) is the decay rate as a decimal
• \( t \) signifies the number of time periods that have passed

This formula provides a straightforward method to calculate the remaining quantity after a certain period of decay. First, determine the decay portion \( 1 - r \), then raise this value to the power \( t \), representing the passage of time. The formula highlights how exponential functions can succinctly describe dynamically changing quantities through time.

Algebraic Calculations

Algebraic calculations often require step-by-step attention to detail to ensure accuracy, especially in exponential decay problems. Using the exponential decay formula involves several essential steps:

• Start with converting percentages to decimals for calculations.
• Simplify expressions inside parenthesis before proceeding to powers or exponents.
• Use a calculator for precise computation of powers like \( 0.84^{11} \).
• Multiply the computed power by the initial amount to find the final value.

Accurate rounding completes the process, especially when instructed to round to the nearest whole number, which often arises in practical calculations. These consistent steps help avoid common errors and are vital for achieving correct results in algebraic problems of exponential decay.