Problem 13 - Algebra 2

Question

For Exercises 13 and \(14,\) use the following information. The annual Gross Domestic Product (GDP) of a country is the value of all of the goods and services produced in the country during a year. During the period \(2001-2001-2004,\) the Gross Domestic Product of the United States grew about \(2.8 \% 1-2001\) per year, measured in 2004 dollars. In \(2001,\) the GDP was \(\$ 9891\) billion. Assuming this rate of growth continues, what will the GDP of the United States be in the year 2015\(?\)

Step-by-Step Solution

Answer

The projected GDP in 2015 is approximately 14,515.25 billion dollars.

Step 1: Understand the Concept of Compound Growth

The GDP grows annually at a rate of 2.8%. This means each year, the GDP increases by 2.8% of its value from the previous year. This type of growth is called compound growth because each year's growth builds on the previous year's total.

Step 2: Write the Formula for Compound Interest

The formula to calculate compound growth over multiple years is similar to compound interest: \[ P = P_0 (1 + r)^t \]Where \( P \) is the future value of the GDP, \( P_0 \) is the initial GDP, \( r \) is the growth rate, and \( t \) is the number of years.

Step 3: Identify Known Values

From the problem, you know: - Initial GDP (\( P_0 \)) = 9891 billion dollars- Annual growth rate (\( r \)) = 0.028 (since 2.8% = 0.028)- Time (\( t \)) = 2015 - 2001 = 14 years

Step 4: Plug Values into the Formula

Now, substitute the known values into the compound growth formula: \[ P = 9891 \times (1 + 0.028)^{14} \]

Step 5: Calculate the Exponent

First, calculate \( (1 + 0.028)^{14} \):\[ (1.028)^{14} \approx 1.4677 \]

Step 6: Calculate the Future GDP

Substitute the calculated exponent back into the formula:\[ P = 9891 \times 1.4677 \approx 14515.25 \]

Step 7: Interpret the Result

Therefore, the projected GDP in 2015, assuming a continuous 2.8% annual growth rate, is approximately 14,515.25 billion dollars.

Key Concepts

Gross Domestic Product (GDP)

The Gross Domestic Product, or GDP, is a crucial concept in economics. It represents the total monetary value of all goods and services produced within a country's borders over a specific time period, typically annually. GDP provides an economic snapshot of a nation, allowing policymakers and analysts to evaluate the health and size of an economy.
Consider this to get a better understanding:

• GDP can reflect the performance of a country's economy - if GDP is increasing, the economy is likely improving.
• It is used to compare the economic progress between different countries.
• GDP influences government policy, investment decisions, and even levels of taxation.

By analyzing GDP, we can infer how well production is managed within a country, and by extension, the wealth and living standards of its population.

compound interest

Compound interest is a powerful financial concept that applies not only to savings and investments but also to growth scenarios like GDP. In the context of GDP, it refers to how the value grows year after year, with each year’s growth being added to the base value for future growth calculations.

• In simple terms, it's earning 'interest on interest'.
• This compound growth means that even small rates of increase can lead to significant changes over time, thanks to the accumulation.

This is why compound interest is often associated with exponential growth, making it a key component in understanding economic growth patterns.

growth rate

The growth rate is a percentage that signifies how much a particular value, such as GDP, increases over a specific period of time. In our example, the U.S. GDP was growing at 2.8% per year.
What you need to know about growth rates:

• A higher growth rate indicates faster economic expansion, while a lower rate suggests slower growth.
• Even small changes in the growth rate can have substantial impacts when compounded over many years.
• Understanding the growth rate is essential for predicting future economic conditions and making informed financial decisions.

The growth rate will tell us how quickly or slowly an economy is expanding, allowing us to forecast its future size and performance.

exponential growth

Exponential growth occurs when a quantity increases at a consistent rate over time, multiplying rather than adding to its previous value. In the context of compound interest and GDP, it refers to how an initially small growth rate can lead to substantial increases over many years due to compounding.

• The idea of multiplying the growth each period results in a curve that accelerates upwards over time, instead of a straight line.
• This means that the impact of growth can seem slow at first but increases dramatically over larger time spans.

Understanding exponential growth is vital to grasp how small, steady rates of increase can lead to significant economic change, as seen in GDP projections.