Problem 21 - Precalculus

Question

The Gross Domestic Product (GDP) of the US (in billions of dollars) \(t\) years after the year 2000 can be modeled by: $$ G(t)=9743.77 e^{0.0514 t} $$ (a) Find and interpret \(G(0)\). (b) According to the model, what should have been the GDP in \(2007 ?\) In \(2010 ?\) (According to the US Department of Commerce, the 2007 GDP was \(\$ 14,369.1\) billion and the 2010 GDP was \(\$ 14,657.8\) billion.)

Step-by-Step Solution

Answer

(a) \( G(0) = 9743.77 \) billion dollars, the GDP in 2000; (b) Predicted GDP: \( 13961.29 \) billion in 2007, \( 16291.78 \) billion in 2010.

Step 1: Evaluating the GDP at t=0

To find \( G(0) \), we need to substitute \( t = 0 \) in the function \( G(t) = 9743.77 e^{0.0514 t} \). Calculating this gives:\[ G(0) = 9743.77 e^{0.0514 imes 0} = 9743.77 e^0 = 9743.77 \] This means that the GDP at the year 2000 is \( 9743.77 \) billion dollars. Thus, \( G(0) \) represents the GDP in the year 2000.

Step 2: Calculating GDP for the Year 2007

For the year 2007, \( t = 2007 - 2000 = 7 \). Substitute \( t = 7 \) in the equation:\[ G(7) = 9743.77 e^{0.0514 imes 7} \]Calculate the exponential term:\[ e^{0.0514 imes 7} = e^{0.3598} \approx 1.433 \]Therefore:\[ G(7) = 9743.77 imes 1.433 \approx 13961.29 \]So the predicted GDP for 2007 is approximately \( 13961.29 \) billion dollars.

Step 3: Calculating GDP for the Year 2010

For the year 2010, \( t = 2010 - 2000 = 10 \). Substitute \( t = 10 \) in the equation:\[ G(10) = 9743.77 e^{0.0514 imes 10} \]Calculate the exponential term:\[ e^{0.0514 imes 10} = e^{0.514} \approx 1.672 \]Therefore:\[ G(10) = 9743.77 imes 1.672 \approx 16291.78 \]So the predicted GDP for 2010 is approximately \( 16291.78 \) billion dollars.

Key Concepts

Gross Domestic Product (GDP)

Gross Domestic Product, often abbreviated as GDP, is a critical measure of a country's economic performance. It essentially represents the total monetary value of all goods and services produced over a specific time period within a nation. GDP is an important indicator because:

• It measures the economic health of a country.
• It's used to compare the economic growth between different countries.
• It serves as a guide for economic policy.

GDP can be expressed in nominal terms, which include inflation, or real terms, which adjust for inflation. It's vital in understanding the scale and performance of an economy.

Exponential Growth

Exponential growth is a powerful concept often used to describe processes that increase rapidly over time. In mathematics, exponential growth refers to an increase at a consistent percentage rate over a period. This can be represented through the formula:\[ P(t) = P_0 \times e^{rt} \]where:

• \( P(t) \) is the amount at time \( t \).
• \( P_0 \) is the initial amount.
• \( r \) is the growth rate.
• \( e \) is the base of the natural logarithm, approximately equal to 2.718.

Exponential growth is significant because:

• It models various real-world processes like population growth, spread of diseases, and economic growth.
• It emphasizes how quickly things can expand or grow if the rate is sustained over time.

In the context of GDP, exponential growth highlights how an economy might rapidly expand, provided it maintains a steady growth rate.

Mathematical Modeling

Mathematical modeling involves using mathematical equations and representations to describe and analyze real-world phenomena. It allows us to predict outcomes and understand relationships between various elements in a system. Key aspects of mathematical modeling include:

• Providing a simplified representation of complex systems.
• Offering a precise way to test hypotheses and predict future events.
• Being essential in fields like economics, physics, biology, and engineering.

For GDP modeling, a mathematical approach helps:

• Project future economic growth.
• Analyze historical trends and compare different economies.
• Inform policy-making and economic strategies.

A well-constructed mathematical model is invaluable for planning and investment decisions.

Yearly GDP Calculation

Yearly GDP Calculation is the process of determining the GDP of a country for a specific year using a model or historical data. In our exercise, we used an exponential growth model to calculate GDP for different years. The process generally follows these steps:

• Identifying the start value, like GDP at a specific base year.
• Using the growth model, substituting the year into the model equation.
• Computing the growth factor using the exponential part of the equation.
• Multiplying the base value by the growth factor to find the projected GDP.

For example, to compute the GDP for the year 2007, we substitute the value for that year into our model, calculate the necessary exponential factor, and multiply it with the GDP from the base year. This process helps us see projected economic trends and compare them with actual data.