Problem 39 - Algebra 2

Question

Multiple choice The gross domestic product, or GDP, of the United States was about \(\$ 10.99\) trillion in 2003 . Assume that GDP grows 3.1\(\%\) each year. Use an exponential model to find which percent best describes the GDP growth that occurs over 18 years. a. 19.0\(\%\) b. 74.7\(\%\) c. 129.1\(\%\) d. 173.2\(\%\)

Step-by-Step Solution

Answer

Therefore, none of the provided answers perfectly match the calculated value. However, 129.1% (Answer c) would be the closest choice to the calculated GDP growth over 18 years.

Step 1: Identify P, r, and n

In this exercise, \( P = 10.99 \) (in trillions), the initial GDP. \( r = 0.031 \), the annual growth rate in decimal form. \( n = 18 \), the number of years.

Step 2: Use the Exponential Growth Formula

Plug \( P \), \( r \), and \( n \) into the formula \( A = P(1 + r)^{n} \) to find \( A \), the value of the GDP after 18 years. Doing so gives \[ A = 10.99 (1 + 0.031)^{18} \].

Step 3: Calculate the GDP After 18 Years

Calculate the value in the bracket first, then raise it to the power of 18, and finally multiply by 10.99. Doing so yields \( A ≈ 25.832 \) trillion.

Step 4: Calculate the Percentage Increase

Finally, calculate the percentage increase, which is the final value minus the initial value, divided by the initial value, and all multiplied by 100. This results in \[ \text{Percentage increase}= \frac{(A - P)}{P} * 100 = \frac{(25.832 - 10.99)}{10.99} * 100 ≈ 134.8 \% \].

Key Concepts

GDP

The term GDP, or Gross Domestic Product, refers to the total monetary value of all goods and services produced in a country over a specific period. It is a critical indicator of a country's economic health and growth. In the exercise example, the GDP of the United States in 2003 was approximately $10.99 trillion. This value serves as a baseline for calculating future growth.
GDP itself can be measured in three different ways:

• Production Method: Measures the output of goods and services.
• Income Method: Calculates the total income earned by residents.
• Expenditure Method: Takes into account the total spending.

Each method should theoretically give the same number for GDP. In the context of this exercise, GDP acts as the starting value in our exponential growth calculations.

Percentage Increase

Percentage increase is a way of expressing the amount of increase, relative to the original size. It helps in understanding how much a value has grown over time compared to its starting point.
In the context of exponential growth, it allows us to track growth over multiple periods. To calculate the percentage increase for the GDP in our example, the process involved finding out how much the GDP grew from its starting value of $10.99 trillion in 2003 to its final value after 18 years.

• Difference: Subtract the initial GDP from the final GDP.
• Divide by Initial: Divide this difference by the initial GDP to find out how much bigger it has become in proportion.
• Convert to Percentage: Multiply this figure by 100 to see it as a percentage.

Understanding percentage increase is essential for making sense of changes in economic conditions over time.

Exponential Model

An exponential model is used to represent situations where growth or decay occurs at a constant percentage rate over time. This is a natural model for many real-world phenomena, such as population growth, radioactive decay, and of course, GDP growth.
In our exercise, the model used is given by the formula:\[ A = P(1 + r)^n \]- **\(A\):** Final amount after time \(n\)- **\(P\):** Initial amount- **\(r\):** Growth rate per period- **\(n\):** Number of periods
By inserting our specific values — initial GDP, the growth rate, and the number of years — into this formula, we can predict or back-calculate the GDP of a nation at any given time. Exponential models need accurate input values, as even small changes in rates or time periods can significantly affect the outcome.

Exponential Growth Formula

The exponential growth formula is crucial in calculating how values grow over time when the rate of growth is proportional to the value itself. This form of growth can be found in various disciplines from finance to natural sciences.
The formula used in this exercise is:\[ A = P(1 + r)^n \]This formula is powerful because it allows us to predict future values based on current data.

• **\(P\) (Principal):** This is the starting value or initial amount, which, in this exercise, is the 2003 GDP.
• **\(r\) (Rate):** Growth rate expressed as a decimal. A 3.1% growth rate becomes 0.031 in the formula.
• **\(n\) (Number of Periods):** The total time over which growth is considered, which is 18 years here.
• **\(A\):** Final amount, which represents GDP after the 18 years.

This formula helps establish a clear and precise method for future economic forecasting, making it a cornerstone of financial and economic models.