Problem 5 - Intermediate Algebra

Question

Practice using the exponential growth formula by completing the table below. Round final amounts to the nearest whole. See Example 1. $$ \begin{array}{|c|c|c|c|} \hline \text {Original } & {\text {Growth Rate per}} & {\text {Number}} & {\text {Final A mount after } x \text {Years}} \\ {\text {Amount}} & {\text {Year}} & {\text {of Years, } x} & {\text {of Growth}} \\ \hline 17 & {29 \%} & {28} \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The final amount after 28 years is 1313.

Step 1: Understand the Exponential Growth Formula

The exponential growth formula is given by \( A = P(1 + r)^t \), where \( A \) is the final amount, \( P \) is the initial amount, \( r \) is the growth rate per period, and \( t \) is the number of periods.

Step 2: Identify the Given Values

From the table, we have the original amount \( P = 17 \), the growth rate per year \( r = 29\% \) or \( 0.29 \) in decimal, and the number of years \( t = 28 \).

Step 3: Substitute the Values Into the Formula

Substitute the values into the exponential growth formula: \( A = 17(1 + 0.29)^{28} \).

Step 4: Calculate the Growth Factor

Calculate the growth factor \( 1 + r \) which is \( 1 + 0.29 = 1.29 \).

Step 5: Compute the Exponentiation

Calculate \( 1.29^{28} \). Use a calculator to find that \( 1.29^{28} \approx 77.205 \).

Step 6: Determine the Final Amount

Multiply the original amount by the calculated growth factor: \( A = 17 \times 77.205 \approx 1313.485 \).

Step 7: Round to the Nearest Whole Number

Round \( 1313.485 \) to the nearest whole number, which is \( 1313 \).

Step 8: Fill in the Table

The final amount after 28 years is \( 1313 \), which completes the table.

Key Concepts

Exponential Growth Formula

Exponential growth describes how a quantity increases over time at a constant rate. This is a common way to model phenomena in finance, biology, and physics. A key tool in understanding this concept is the Exponential Growth Formula. The formula is expressed as:\[ A = P(1 + r)^t \]where:- \( A \) is the final amount after growth.- \( P \) is the initial principal amount.- \( r \) is the growth rate per period (expressed as a decimal).- \( t \) is the number of periods.This formula allows us to calculate how much a starting quantity grows after several periods at a specific rate. It's important to express the growth rate \( r \) as a decimal rather than a percentage, so remember to divide by 100 if starting from a percentage. In practice, whether it’s populations doubling or investments growing, this formula is invaluable for making predictions about the future.

Rounding Numbers

Rounding numbers is an essential mathematical skill, especially when dealing with figures that have a lot of decimal places. It helps to simplify results and make them more understandable. In our context, after solving the exponential growth formula, we often end up with numbers that are not whole.Here's how rounding works:- Look at the digit to the right of where you want to round.- If this digit is 5 or greater, increase the rounding digit by 1.- If it's less than 5, leave the rounding digit unchanged.For example, after calculating an exponential value, you might get \( 1313.485 \). Since the first digit after the decimal point is 4 (less than 5), the number rounds down to \( 1313 \). Rounding simplifies complex numbers and makes results in real-world applications more manageable.

Growth Rate

Understanding the growth rate is crucial in applying the exponential growth formula. It represents how much the initial amount increases in one period. In the formula, the growth rate \( r \) must be converted from a percentage to a decimal by dividing by 100. For example, a growth rate of 29% becomes 0.29 in the calculation.The growth rate shows:- How steeply the number is expected to grow.- The velocity at which the principal amount changes over time.To make predictions using the growth rate, it's crucial to ensure that it's correctly converted to a decimal. This then affects the overall exponential calculation, determining how much the initial quantity will increase over the specified periods.

Calculations with Exponents

Exponents are used to express repeated multiplication of a number by itself. Calculations with exponents are fundamental to computing exponential growth. The expression \((1 + r)^t\) shows that we repeatedly multiply the base, \(1 + r\), \(t\) times.In exponential growth problems:- The exponent \( t \) represents the number of periods the growth process is observed over.- Calculate the base \(1 + r\) for each period.For instance, calculating \(1.29^{28}\) involves using a calculator for accuracy because manual multiplication would be cumbersome. Exponents compress complex multiplications into manageable calculations, showing significant growth over time. This approach helps us predict and understand growth trends powerfully and succinctly.