Problem 7 - Algebra A Combined Function

Question

Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Decay Rate } \\ \text { per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Decay } \end{array} \\ \hline 305 & 5 \% & 8 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The final amount after 8 years of decay is 203.

Step 1: Understand the Exponential Decay Formula

The formula for exponential decay is given by \( A = P(1 - r)^x \), where \( A \) is the final amount, \( P \) is the original amount, \( r \) is the decay rate, and \( x \) is the number of years. In this exercise, you will apply this formula to find the final amount after a period of decay.

Step 2: Identify Given Values

From the table, the original amount \( P \) is 305, the decay rate \( r \) is 5\% (or 0.05 as a decimal), and the number of years \( x \) is 8. These values will be substituted into the exponential decay formula.

Step 3: Substitute into the Formula

Substitute the identified values into the exponential decay formula: \( A = 305(1 - 0.05)^8 \). This simplifies to \( A = 305(0.95)^8 \).

Step 4: Calculate the Decay Factor

Calculate \( (0.95)^8 \). Doing this calculation, we find \( (0.95)^8 \approx 0.6634 \).

Step 5: Compute the Final Amount

Multiply the decay factor by the original amount to find the final amount: \( A = 305 \times 0.6634 \). This equals approximately \( 202.627 \).

Step 6: Round the Final Amount

Round the computed final amount \( 202.627 \) to the nearest whole number, which is 203.

Key Concepts

exponential decay formula

Exponential decay describes the process where a quantity decreases at a consistent percentage rate over time. To express this mathematically, we use the exponential decay formula, which is expressed as \( A = P(1 - r)^x \). Here, \( A \) denotes the final amount after the decay has taken place, \( P \) is the original amount you started with, \( r \) is the decay rate, and \( x \) represents the time period in years.
This formula helps in modeling various real-world scenarios like radioactive decay, depreciation of assets, and more.
To use the formula effectively, you substitute the given values into the equation. Understanding each component and how they interact is crucial for solving problems using this formula.

decay rate

The decay rate is a critical factor in determining how quickly something diminishes over time. It is represented by \( r \) in the exponential decay formula and is typically expressed as a percentage. In our exercise, the decay rate given is 5%.
To simplify it for use in the formula, you need to convert this percentage into a decimal form, which would be 0.05. This conversion ensures that the calculations performed within the formula work correctly.
Analyzing the decay rate can provide insights into how rapidly changes occur in the system you are studying, whether it’s a substance decreasing over time or a financial asset losing its value.

original amount

The original amount, denoted by \( P \) in the exponential decay formula, is the starting point from which the decay process begins. In the context of our exercise, it is given that \( P = 305 \).
This is the value before any decrease due to decay is applied. The original amount is crucial because it serves as the baseline for calculating how much of the quantity remains after a set period of time.
Understanding the original amount enables accurate computation of the decay process and helps in projecting future values or states based on the given decay rate.

final amount calculation

Calculating the final amount involves applying the exponential decay formula to determine what remains after a certain period. With all variables known, such as the original amount, decay rate, and time period, you substitute them into the formula \( A = P(1 - r)^x \).
For our task, the final amount is computed by substituting \( P = 305 \), \( r = 0.05 \), and \( x = 8 \) years into the formula:

• First, calculate the decay factor \((1 - r)^x\), which becomes \((0.95)^8\).
• Doing the math, \((0.95)^8\) equals approximately 0.6634.
• Then, multiply this by the original amount: \(305 \times 0.6634\).
• This results in a calculated value of about 202.627.

To conclude, rounding to the nearest whole number gives a final amount of 203. This step shows you how the exponential decay impacts the original amount over the specified period.