Problem 8 - Algebra A Combined Function

Question

Practice using the exponential decay formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Decay Rate } \\ \text { per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Decay } \end{array} \\ \hline 402 & 7 \% & 5 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The final amount after 5 years of decay is 280.

Step 1: Understand the Formula

The exponential decay formula is given by: \[ A = A_0 (1 - r)^x \] Where, \( A_0 \) is the original amount, \( r \) is the decay rate as a decimal, and \( x \) is the number of years.

Step 2: Convert Decay Rate

The decay rate is given as 7%. To use it in our formula, we need to convert it to a decimal. \[ r = \frac{7}{100} = 0.07 \]

Step 3: Substitute Given Values

Now, substitute the given values into the formula: \( A_0 = 402 \), \( r = 0.07 \), \( x = 5 \). Thus, \[ A = 402 \times (1 - 0.07)^5 \]

Step 4: Calculate Intermediate Exponent

Calculate \( (1 - 0.07)^5 \): \[ 1 - 0.07 = 0.93 \] Then, \[ 0.93^5 \approx 0.6983372969 \]

Step 5: Find Final Amount

Multiply the intermediate result by the original amount: \[ A = 402 \times 0.6983372969 \approx 280.34038 \] Round this amount to the nearest whole number: Final amount = 280.

Key Concepts

Exponential Decay Formula

The exponential decay formula is a powerful tool used to calculate the reduction of an initial amount over time. This formula is essential in fields such as finance, physics, and biology, where the concept of decay is often applicable. It is expressed as: \[ A = A_0 (1 - r)^x \]- \( A \) represents the final amount after decay.- \( A_0 \) is the original amount before decay.- \( r \) is the decay rate (expressed as a decimal).- \( x \) is the number of years the decay has occurred.To use this formula effectively, it is crucial to convert the decay rate from a percentage to a decimal. This conversion ensures accurate calculations. Understanding this formula allows you to predict how quickly an amount decreases over time, aiding in various real-world applications.

Decay Rate

The decay rate indicates how quickly an amount decreases over a set period. It is typically expressed as a percentage. However, to be used in an exponential decay formula, it must be converted into a decimal. Conversion from percentage to decimal:

• Divide the percentage value by 100.
• For example, a 7% decay rate becomes 0.07.

The decimal form of the decay rate is then used to compute the extent of decay over time. A smaller decay rate means a slower decrease, while a higher decay rate indicates a quicker reduction. Understanding the decay rate is crucial in predicting the future value of an original amount after ongoing decay.

Rounding Numbers

Rounding numbers is an essential skill, especially when dealing with calculations that involve multiple decimal places. It simplifies numbers to a form that is easier to interpret and useful in real-world applications. Basic steps for rounding:

• Identify the place to which you want to round.
• Look at the digit immediately to the right of this place.
• If that digit is 5 or greater, increase the rounding digit by one.
• If it's less than 5, keep the rounding digit the same.

In the context of exponential decay, rounding helps in delivering answers that are practical and realistic, as seen when 280.34038 is rounded to 280. This avoids overly complex numbers which might not be necessary in everyday scenarios.

Algebraic Functions

Algebraic functions are expressions that involve variables, constants, and algebraic operations such as addition, subtraction, multiplication, and division. They form the backbone of mathematical problem-solving and are fundamental in understanding decay models.In exponential decay scenarios, functions describe the relationship between variables such as time, decay rate, and amount. The formula \( A = A_0 (1 - r)^x \) is an example of an algebraic function that models how quantities diminish over time.These functions allow us to solve for unknown variables and predict future outcomes based on given data. They play a crucial role in developing strategies and solutions in scientific, engineering, and financial contexts.