SOLUTION: The number of automobiles in the United States in 1900 was around 8000. In the year 2000, the number of automobiles in the United States reached 200 million. Find an exponential mo

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Question 136215This question is from textbook Mathematical Excursions
: The number of automobiles in the United States in 1900 was around 8000. In the year 2000, the number of automobiles in the United States reached 200 million. Find an exponential model for the data and use the model to predict the number of automobiles in the United Sates in 2010. Use t=0 to represent the year 1900. Round to the nearest hundred thousand automobiles. This question is from textbook Mathematical Excursions

Answer by ankor@dixie-net.com(22740) About Me  (Show Source):
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The number of automobiles in the United States in 1900 was around 8000. In the year 2000, the number of automobiles in the United States reached 200 million. Find an exponential model for the data and use the model to predict the number of automobiles in the United Sates in 2010. Use t=0 to represent the year 1900. Round to the nearest hundred thousand automobiles.
:
Let y = no. of cars
:
y = .08(a^t)
:
Find the value of a when t = 100 (yrs) & y = 200,000,000/100,000 = 2000 (100k cars)
:
.08(a^100) = 2000
:
ln(.08) + 100*ln(a) = ln(2000); use nat logs, and log equiv of exponents
:
-2.5257 + 100*ln(a) = 7.6
:
100*ln(a) = 7.6 + 2.5257
:
ln(a) = 10.1266%2F100
ln(a) = .101266
a = 1.10657
:
Use the formula
y = .08(1.10657^t)
:
Find y in 2010; t = 110
y = .08(1.10657^110)
y = .08 * 68814.2
y = 5505 which is: 5505 * 100,000 = 550,500,000 cars in 2010