Question

In 1995, the Gross Domestic Product (GDP) of the United States was $7,398 billion. In 2000, it was $9,817 billion.

(a) Develop the exponential model that represents the nation's GDP. (Write your model in terms of t, where t is the number of years after 1995. Let p represent the GDP in billions. Round the coefficient of t to seven decimal places.)

(b) Use the model to predict the GDP in 2004. (The actual 2004 GDP was $11,735 billion. Round your answer to the nearest whole number.)

Answer 1

Given

p(0) = 7398

p(5) = 9817

p(t) is the predicted GDP in billions in t years after 1995.

Find

a) an exponential model for p(t)

b) p(9)

Solution

There are a number of ways to write exponential models. My favorite uses the given numbers exactly, so no rounding or approximation is necessary. This version gives

... p(t) = 7398·(9817/7398)^(t/5)

Then

... p(9) = 7398·(9817/7398)^(9/5) ≈ 12,310

If you want to use e as the base of your exponential function, then the exponent multiplier in e^(kt) becomes

... k = ln(9817/7398)^(1/5) ≈ 0.0565812

so the function is

... p(t) = 7398·e^(0.0565812t)

and

... p(9) ≈ 12,310

The GDP in 2004 is predicted to be 12,310 billion.

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[editorial comment] It is not surprising that the estimate is a little high. The model assumes annual growth of 5.8%, which is a little higher than the long-term average. Over the 9 years from 1995 to 2004, the average was actually 5.26%