Question

The economy of Argentina as measured by its Gross Domestic Product (GDP) is shrinking at a rate of \(2.6 \%\) per year. In 2015 , the GDP of Argentina was \(\$ 630\) billion. Which of the following functions represents Argentina's GDP, \(A\), in billions of dollars, \(y\) years since \(2015 ?\) A) \(A(y)=630-(1-0.26) y\) B) \(A(y)=630(1-0.26)^y\) C) \(A(y)=630-(1-0.026) y\) D) \(A(y)=630(1-0.026)^y\)

Short answer

The function that represents Argentina's GDP in billions of dollars, \(y\) years since 2015, is: \(A(y)=630(1-0.026)^y\)

Step-by-step solution

Identify exponential decay functions

Between the given options, we can identify that both option B and option D represent exponential decay functions because they have the form \(A(y)=630(1-r)^y\), where \(r\) is the decay rate.

Convert the decay rate

We have been given a decay rate of \(2.6\%\) per year. To use this percentage in the function, we need to convert it to a decimal. We can do that by dividing by 100: \(2.6\% = 2.6 / 100 = 0.026\)

Compare the options

Now that we have the decay rate as a decimal, we can compare our options to see which function accurately represents Argentina's GDP: Option B: \(A(y)=630(1-0.26)^y\) In this option, the decay rate is 0.26, which is not equal to our calculated decay rate of 0.026. Option D: \(A(y)=630(1-0.026)^y\) In this option, the decay rate is 0.026, which is equal to our calculated decay rate.

Conclude the answer

Since option D has the exponential decay format and the correct decay rate, we can conclude that the function that represents Argentina's GDP in billions of dollars, \(y\) years since 2015, is: \(A(y)=630(1-0.026)^y\)

Key concepts

GDP calculation

Gross Domestic Product, commonly known as GDP, is a crucial indicator used to gauge the health of a country's economy. It represents the total market value of all final goods and services produced within a country's borders over a specific period, usually a year. GDP can be used to compare the economic performance of different countries, or to track the economic progress of a single country over time.
When measuring GDP, various components are considered, including consumer spending, business investment, government spending, and net exports (the value of exports minus the value of imports). Understanding changes in GDP is essential for economic planning and policy making, as it offers insights into economic growth or decline.
In the case of Argentina's GDP, the focus is on the decline over time, which is represented by a shrinking GDP. This decline can be captured and analyzed using exponential functions, helping economists to anticipate future economic conditions.

decay rate conversion

To analyze economic shrinkage, like the GDP decline of Argentina, it's vital to understand the concept of decay rate. The decay rate is essentially the percentage by which a quantity decreases annually. In Argentina’s case, the GDP is shrinking at a rate of 2.6% per year.
However, for mathematical calculations, it's essential to convert this percentage into a decimal form. This is done by dividing the percentage by 100. So, a 2.6% decay rate becomes 0.026 in decimal form. This conversion is crucial because it allows us to utilize decay rates in mathematical models and exponential functions effectively.
Correctly converting decay rates ensures accurate representation of the economic decline in mathematical terms, allowing for clearer predictions and understanding of future trends.

mathematical modeling

Mathematical modeling serves as a powerful tool in representing and forecasting real-world scenarios using mathematical expressions. In the context of Argentina's GDP, a mathematical model can be used to predict future GDP values given the current rate of decline.
The model typically involves the construction of an equation using known values, like the initial GDP and the decay rate. These values are applied in an exponential decay function, which systematically outlines how the GDP is expected to decrease annually. Mathematical models play a critical role in helping economists and policymakers visualize and plan for economic changes.
By using mathematical models, we can anticipate how long it might take for a GDP to shrink to a specific level, allowing individuals and governments to make informed financial decisions.

exponential functions

Exponential functions are mathematical expressions that depict how a quantity changes exponentially over time. They are key in modeling scenarios where growth or decline happens at a constant percent rate, like Argentina’s GDP decline of 2.6% per year.
In an exponential decay function, the formula takes the form \(A(t) = A_0(1-r)^t\), where \(A_0\) is the initial quantity (Argentina's GDP in 2015, in this case, $630 billion), \(r\) is the decay rate (0.026 after conversion), and \(t\) represents time in years. As time progresses, the GDP decreases exponentially, which is visibly represented by the function.
Exponential functions provide a clear and concise way to understand how different economic variables, like GDP, can decrease over time, providing significant insights for economic analysis.