Population Aging The following table shows the percentage of U.S. residents over the age of 85 in 1950, 1960, . . . , 2010 (t is time in years since 1900):
a. Find the logarithmic regression model of the form P(t) = A ln t + C. (Round the coefficients to four significant digits). HINT [See Example 5.]
b. In 2020, 2.1% of the population is projected to be over 85. To how many significant digits does the model reflect this figure?
c. Which of the following is correct? If you increase A by 0.1 and decrease C by 0.1 in the logarithmic model, then
(A) The new model predicts eventually lower percentages.
(B) The long-term prediction is essentially the same.
(C) The new model predicts eventually higher percentages.
Example 5:
The following table shows the total spent on research and development in the United States, in billions of dollars, for the period 1995–2009 (t is the year since 1990).*
Find the best-fit logarithmic model of the form
S(t) = A ln t + C
and use the model to project total spending on research in 2012, assuming the trend continues.
Solution We use technology to get the following regression model:
S(t) = 83.01 ln t + 64.42. Coefficients rounded
Because 2012 is represented by t = 22, we have
S(22) = 83.01 ln(22) + 64.42 ≈ 320. Why did we round the result to two significant digits?
So, research and development spending is projected to be around $321 billion in 2012.
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