Problem 10 - Algebra 2

Question

MONEY For Exercises 10 and 11 , use the following information. In \(1993,\) My- Lien inherited \(\$ 1,000,000\) from her grandmother. She invested all of the money, and by 2005 , the amount had grown to \(\$ 1,678,000\) . Write an exponential function that could be used to model the money \(y .\) Write the function in terms of \(x,\) the number of years since \(1993 .\)

Step-by-Step Solution

Answer

The exponential function is \( y = 1000000(1.044)^x \).

Step 1: Identify Initial and Final Amounts

The initial amount invested by My-Lien from the inheritance is \( P = \\(1,000,000 \). The final amount in 2005 is \( A = \\)1,678,000 \). We need to express this relationship using an exponential growth function.

Step 2: Determine Time Duration

Calculate the duration between 1993 and 2005. The number of years between 1993 and 2005 is \( x = 2005 - 1993 = 12 \) years.

Step 3: Formulate Exponential Growth Model

The general form of the exponential growth model is \( y = P e^{rx} \) where \( r \) is the rate of growth, \( P \) is the initial principal amount, and \( x \) is the number of years. However, to make it simpler, we can use the formula \( A = P(1 + r)^x \). We need to find \( r \).

Step 4: Solve for Growth Rate

Using the formula \( A = P(1 + r)^x \), substitute \( A = 1678000 \), \( P = 1000000 \), and \( x = 12 \):\[ 1678000 = 1000000(1 + r)^{12} \]Divide both sides by 1000000:\[ 1.678 = (1 + r)^{12} \]

Step 5: Use Algebra to Find r

Take the 12th root on both sides to solve for \( 1 + r \):\[ 1 + r = 1.678^{\frac{1}{12}} \]

Step 6: Simplify to Find r

Calculate to find \( r \):\[ r \approx 1.678^{\frac{1}{12}} - 1 \cong 0.044 \]Thus, \( r = 4.4\% \).

Step 7: Write Exponential Function

Now plug the values back into the exponential function formula \( y = P(1 + r)^x \):\[ y = 1000000(1 + 0.044)^x \]Simplify to get:\[ y = 1000000(1.044)^x \]

Key Concepts

Initial Amount

In any exponential growth scenario, understanding the 'initial amount' is crucial. Think of this as your starting point—the very foundation of growth.
In the context of the given exercise, the initial amount is the money My-Lien received from her grandmother. This sum was \( \$1,000,000 \).
The initial amount is often represented by the variable \( P \) in mathematical formulas.

• It is the principal or starting investment before any growth occurs.
• The initial amount serves as a benchmark to measure how much growth or change has happened over time.

Recognizing this concept helps us establish the baseline from which all future calculations stem. In the exponential growth function, it's the number you're multiplying and growing exponentially.

Final Amount

The 'final amount' is what you end up with after some time, following growth or changes.
In My-Lien's case, by 2005, her initial \( \\(1,000,000 \) had grown to a final amount of \( \\)1,678,000 \).
This figure tells us the total effect of the applied growth over the period.The representation of final amount in an exponential equation is usually denoted by \( A \).

• Final amount illustrates the achieved result after multiplication of growth factors over time.
• It helps in calculating or confirming the rate of growth that has taken place.

Thus, knowing the final amount allows us to measure the effectiveness or success of the growth pattern under question.

Growth Rate

The 'growth rate' is a crucial component of calculating exponential growth.
It's a measure of how quickly something is increasing over time and is usually expressed as a percentage.
In the problem you're working with, the growth rate \( r \) was found to be 4.4%. This value was calculated using a combination of the initial amount, the final amount, and the time span over which the growth occurred.

• Growth rate signifies the percentage change over each time period. It helps in predicting future values.
• In mathematical terms, it shows how much of the previous amount is added to itself to get the new amount.

Calculating the growth rate involves solving for \( r \) in the model of \( A = P(1 + r)^x \), allowing predictions of future growth.

Exponential Model

The 'exponential model' is a mathematical representation of how a quantity grows over time.
It provides a formula that describes exponential growth, which is characterized by rates that continuously increase over time.
In this case, My-Lien's example is structured as \( y = 1000000(1.044)^x \). Here:

• \( y \) represents the amount of money after \( x \) years from 1993.
• \( P = 1000000 \) is the initial amount, \( 1.044 \) is representative of adding the 4.4% growth rate, and \( x \) is the time in years.

The exponential model is a powerful tool for analyzing how investments grow or how populations change under consistent growth conditions. By manipulating the model, one can predict future scenarios or understand previous growth behaviors accurately.