Problem 1 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The final amount after 8 years is 451.

Step 1: Identify the Exponential Growth Formula

The formula for exponential growth is \( A = P(1 + r)^x \), where \( A \) is the final amount, \( P \) is the original amount, \( r \) is the growth rate per period (expressed as a decimal), and \( x \) is the number of periods.

Step 2: Plug in Known Values

Given: Original Amount \( P = 305 \), Growth Rate \( r = 5\% = 0.05 \), Number of Years \( x = 8 \). Substitute these values into the formula: \( A = 305(1 + 0.05)^8 \).

Step 3: Simplify the Formula

Simplify the expressions inside the parentheses: \( 1 + 0.05 = 1.05 \). This updates the formula to \( A = 305(1.05)^8 \).

Step 4: Calculate the Exponent

Compute \( (1.05)^8 \) using a calculator to get approximately 1.477455.

Step 5: Find the Final Amount

Multiply the calculated value by the original amount: \( A = 305 \times 1.477455 \approx 450.597 \).

Step 6: Round the Final Amount

Round the final amount to the nearest whole number, which is 451.

Key Concepts

Understanding Growth Rate

The growth rate is a critical factor in understanding how an investment, population, or any quantity increases over time. It represents the proportional increase in size over a given period. Expressed as a percentage, it quantifies how rapidly something grows. To use it in calculations involving exponential growth, it's essential to convert the percentage into a decimal format. For instance, a growth rate of 5% becomes 0.05 when expressed in decimals. Converting percentages into decimals involves dividing the percentage value by 100. This step is crucial because it aligns the growth rate's format with the requirements of mathematical formulas, such as the exponential growth formula.

Key points to remember:

• The growth rate indicates how quickly a value increases.
• Convert the percentage into a decimal for calculations.
• It is commonly denoted by the symbol \( r \) in formulas.

The Concept of Original Amount

When discussing exponential growth, the term 'original amount' is foundational. It refers to the initial quantity or value before any growth occurs. This might be the initial population number, the starting amount of money in an account, or the baseline level of any increasing parameter. Understanding the original amount is vital because it serves as the baseline from which growth is measured. In formulas, it is usually represented by \( P \) (standing for principal when discussing financial applications).

The original amount is crucial because:

• It serves as the starting point in growth calculations.
• Represents the value at time zero in growth scenarios.
• Is multiplied by the growth factor to determine the future value.

Knowing this helps in accurately predicting how much something will grow over time when applying the exponential growth formula.

The Exponential Growth Formula

The exponential growth formula is a mathematical expression used to calculate how much a quantity will grow over a given number of time periods, at a consistent growth rate. The formula is given by \( A = P(1 + r)^x \), where:

• \( A \) is the final amount after growth.
• \( P \) is the original amount (starting value).
• \( r \) is the growth rate per period (expressed as a decimal).
• \( x \) is the number of periods the growth will occur.

The formula works by repeatedly multiplying the original amount by a growth factor across the specified number of periods. The growth factor, \( (1 + r) \), accounts for both the original amount and the increase associated with each period. As \( x \) increases, \( (1 + r)^x \) expands the amount, illustrating the power of exponential growth.

This powerful formula is essential for:

• Calculating future values based on current assets and growth assumptions.
• Understanding the compounding effect of growth on different quantities over time.
• Predicting long-term trends in various fields, such as finance, biology, and physics.

Applying the exponential growth formula allows for precise predictions and strategic planning based on potential future scenarios.