Problem 6 - Algebra A Combined Function

Question

Practice using the exponential growth formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Growth } \\ \text { Rate per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Growth } \end{array} \\ \hline 29 & 61 \% & 12 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The final amount after 12 years is roughly 7337.

Step 1: Understand the Formula

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

Step 2: Convert Growth Rate to Decimal

The growth rate given is 61%. To convert this percentage to a decimal, divide by 100:\[r = \frac{61}{100} = 0.61\]

Step 3: Substitute the Values into the Formula

Insert the given values into the formula:- \(P = 29\)- \(r = 0.61\)- \(x = 12\)Now the formula becomes:\[A = 29(1 + 0.61)^{12}\]

Step 4: Calculate the Base

First, calculate the base \((1 + r)\): \[1 + 0.61 = 1.61\]

Step 5: Calculate the Exponent

Raise the base to the power of \(x = 12\):\[1.61^{12}\]

Step 6: Calculate the Exponential Growth

Calculate \(1.61^{12}\), then multiply by 29:\[A = 29 \times 1.61^{12} \]\[A \approx 29 \times 252.99 \]\[A \approx 7336.71\]

Step 7: Round the Final Amount

Round \(7336.71\) to the nearest whole number. This gives:\[A \approx 7337\]

Key Concepts

Exponential Growth Formula

Exponential growth is a powerful concept in mathematics used to model situations where a quantity increases at a rate proportional to its current value. The general formula for exponential growth is \[ A = P(1 + r)^x \], where:

• \( A \) is the final amount after growth,
• \( P \) is the initial or original amount,
• \( r \) is the growth rate, expressed as a decimal, and
• \( x \) is the number of time periods the growth occurs for, often years.

The key idea here is that exponential growth covers phenomena where growth compounds on itself, similar to how interest compounds in a bank account. Start by identifying each part of the formula, so you understand how to set up and solve exponential calculations lay effortlessly.

Growth Rate Conversion

Converting the growth rate from a percentage to a decimal form is key in applying the exponential growth formula effectively. Suppose you have a growth rate defined as a percentage like 61%. Conversion involves dividing by 100 to move the decimal point two places to the left:

• Divide 61 by 100 to get \( 0.61 \).

This decimal form of the growth rate is what you use in the formula. The conversion simplifies mathematical operations, allowing us to accurately calculate exponential growth. Keep this process in mind whenever you encounter percentages in calculations.

Exponentiation

Exponentiation is the process of raising a number to the power of another number. In the context of exponential growth, exponentiation is used to find how a quantity grows over a set period through repeated multiplication. Given a base number \(1 + r\), such as \(1.61\), and a number of periods \(x\), for example \(12\) years, the expression becomes \((1.61)^{12}\).
Using a calculator:

• Calculate \(1.61^{12}\).

Exponentiation involves multiplying \(1.61\) by itself 12 times. It represents the factor by which the original amount multiplies after 12 years of growth at 61% per year. Understanding this step gives clarity on how repeated growth affects an amount over time.

Rounding Numbers

Once we have our final calculated amount, precision in figures is important. However, in many practical cases, we round numbers to make them easier to understand or report. Rounding involves adjusting a number to the nearest specified place value. Here, rounding involves converting \(7336.71\) to the nearest whole number:

• Look at the first decimal place—7, in this case.
• Since it's 5 or greater, round up the whole number.
• Thus, \(7336.71\) becomes \(7337\).

Rounding makes the amount more manageable while retaining the essential magnitude of growth. It's a simple yet necessary technique to understand in numerical problem-solving.