Population Aging The following table shows the percentage of U.S. resident...

Population Aging The following table shows the percentage of U.S. residents over the age of 65 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 1940, 6.9% of the population was over 65. To how many significant digits does the model reflect this figure?

c. Which of the following is correct? The model, if extrapolated into the indefinite future, predicts that

(A) The percentage of U.S. residents over the age of 65 will increase without bound.

(B) The percentage of U.S. residents over the age of 65 will level off at around 14.2%.

(C) The percentage of U.S. residents over the age of 65 will eventually decrease

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.