Problem 2 - Algebra A Combined Function

Question

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

Step-by-Step Solution

Answer

The final amount after 5 years is 564.

Step 1: Understand the Exponential Growth Formula

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

Step 2: Extract and Convert the Given Values

From the table, the original amount \( P \) is 402, the growth rate \( r \) is 7\% or 0.07 as a decimal, and the number of years \( x \) is 5.

Step 3: Substitute the Values into the Formula

Replace the variables in the formula with the given values: \( A = 402(1 + 0.07)^5 \).

Step 4: Calculate the Base of the Exponential Expression

Calculate \( 1 + 0.07 = 1.07 \).

Step 5: Raise the Base to the Power of the Number of Years

Compute \( 1.07^5 = 1.402551728 \).

Step 6: Multiply by the Original Amount

Calculate \( 402 \times 1.402551728 = 563.826194 \).

Step 7: Round the Final Amount

Round 563.826194 to the nearest whole number, resulting in 564.

Key Concepts

Growth Rate

When we talk about growth rate, we're typically discussing how quickly something increases over time. It's a percentage that shows the rate at which something, like a population or investment, grows. In the context of exponential growth, the growth rate refers to the percentage change per time period. The growth rate helps to determine how fast the original amount will increase. In our exercise, the growth rate is given as 7% per year. However, to use it in the exponential growth formula, we need to express it as a decimal fraction.

• Convert percentage to decimal: 7% becomes 0.07

This decimal form is then used in calculations to determine the compound growth over the specified time period.

Exponential Growth Formula

The exponential growth formula is a mathematical representation used to calculate the increased amount over time when growth happens at a consistent percentage or rate. The formula is: \[ A = P(1 + r)^x \] Here's a breakdown of what each part of this formula represents:

• \( A \): Final amount after the growth period.
• \( P \): Original amount before the growth starts.
• \( r \): Growth rate per period, expressed as a decimal.
• \( x \): The number of growth periods (such as years).

Using this formula, you can calculate how much something grows when compounded over time. It's a powerful tool for scenarios involving finance, biology, or any field that involves compounding growth.

Calculation Steps

To solve the problem using the exponential growth formula, follow these simple steps: First, understand the values from the exercise:

• Original amount \( P \): 402
• Growth rate \( r \): 7% or 0.07 as a decimal
• Number of years \( x \): 5

Next, plug these values into the exponential growth formula: \( A = 402(1 + 0.07)^5 \). Compute the inside of the parenthesis: \( 1 + 0.07 = 1.07 \). Raise this result to the power of 5 to account for the growth over five years: \( 1.07^5 = 1.402551728 \). Finally, multiply the original amount by this factor to get the final amount: \( 402 \times 1.402551728 = 563.826194 \). Round your answer to the nearest whole number: 564. These steps take you through the calculation process ensuring you understand each part of the formula and how they contribute to the final result.