Problem 5 - Algebra A Combined Function

Question

Practice using the exponential growth formula by completing the table below. Round final amounts to the nearest whole. $$ \begin{array}{|c|c|c|c|} \hline \begin{array}{c} \text { Original } \\ \text { Amount } \end{array} & \begin{array}{c} \text { Growth } \\ \text { Rate per Year } \end{array} & \begin{array}{c} \text { Number } \\ \text { of Years, } \boldsymbol{x} \end{array} & \begin{array}{c} \text { Final Amount after } \\ \boldsymbol{x} \text { Years of Growth } \end{array} \\ \hline 17 & 29 \% & 28 & \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The final amount after 28 years is 745.

Step 1: Understand the Exponential Growth Formula

The exponential growth formula is used to calculate the future value of an investment or population after a certain number of years, given an initial amount, a growth rate, and a time period. The formula is given by: \[ A = P(1 + r)^t \]where: \( A \) is the final amount, \( P \) is the original amount, \( r \) is the growth rate per period (as a decimal), and \( t \) is the number of periods.

Step 2: Convert the Growth Rate to Decimal

The problem states a growth rate of 29%. To use this in calculations, convert the percentage to a decimal by dividing by 100. \[ r = \frac{29}{100} = 0.29 \]

Step 3: Apply the Exponential Growth Formula

Substitute the values from the table into the exponential growth formula. Here, \( P = 17 \), \( r = 0.29 \), and \( t = 28 \). \[ A = 17(1 + 0.29)^{28} \]

Step 4: Calculate the Growth Factor

Calculate the growth factor \( 1 + r \): \[ 1 + 0.29 = 1.29 \]

Step 5: Raise the Growth Factor to the Power of Years

Raise the growth factor to the power of the number of years: \[ (1.29)^{28} \approx 43.745 \]

Step 6: Calculate the Final Amount

Multiply the original amount by the growth factor raised to the power, as calculated: \[ A = 17 \times 43.745 \approx 744.665 \]

Step 7: Round the Final Amount

Round the final amount to the nearest whole number. \( 744.665 \) rounded to the nearest whole number is \( 745 \).

Key Concepts

Growth Rate

In the world of mathematics and real-life applications, the growth rate is a vital concept, especially when talking about exponential growth. A growth rate is essentially a percentage that tells you how much an original amount increases over time, usually per period like a year. To actually use this rate in calculations, like in our example, it's vital to convert it to a decimal. For instance, a 29% growth rate becomes 0.29 when divided by 100. This decimal form simplifies calculations and is necessary for using the exponential growth formula.

• First, identify the percentage growth rate given.
• Divide by 100 to convert the percentage to decimal form.
• This decimal is denoted by \( r \) in mathematical formulas.

So, whenever you encounter growth rates, remember this simple conversion step to set the stage for further calculations.

Exponential Growth Formula

The exponential growth formula is your mathematical toolkit for predicting how an amount grows over time. This formula is widely used, from calculating interest on investments to modeling populations. The formula is written as: \[ A = P(1 + r)^t \]Where:

• \( A \) is the final amount after growth, the target of our calculation.
• \( P \) is the original amount you start with.
• \( r \) is the growth rate per period expressed as a decimal.
• \( t \) is the number of periods, like years, you're calculating for.

You can think of this formula as a process. First, you calculate the new growth rate by adding 1 to your decimal growth rate, which forms your growth factor. Next, you raise this growth factor to the power of the number of periods. Effectively, this step shows how the original amount multiplies each period.By methodically following this formula, you can accurately measure projected growth.

Final Amount Calculation

The ultimate goal of using the exponential growth formula is to find the final amount—what your original amount becomes after growth. Calculating this involves a few systematic steps once you have substituted values into the formula. First, calculate the growth factor by adding 1 to the decimal growth rate, as shown in \( 1 + r \). Then, raise this growth factor to the power of the number of periods (years, in most cases). For example, if our growth factor is 1.29 and we calculate for 28 years, it would be \( 1.29^{28} \), showing the exponential effect over time.Finally, multiply this number by the original amount \( P \). For instance, with an original amount of 17, the computation looks like \[ 17 \times 43.745 \approx 744.665 \]. The final step is simple: rounding this amount to the nearest whole number for practical applications, which results in 745.By methodically applying these steps, you can determine how exponential growth affects any quantity over specified periods.