+216 In this exercise we consider data from the Statistical Abstract of the United States on the fraction of women married for the first time in 1960 whose marriage reached a given anniversary number. The data show that the fraction of women who reached their fifth anniversary was 0.928. After that, for each one-year increase in the anniversary number, the fraction reaching that number drops by about 2%. These data describe constant percentage change, so it is reasonable to model the fraction M as an exponential function of the number n of anniversaries since the fifth.
(a) What is the yearly decay factor for the exponential model?
M = ________
(b) Find an exponential model for M as a function of n. (Let n = 0 represent the fifth anniversary.) \
(c) According to your model, what fraction of women married for the first time in 1960 celebrated their 45th anniversary? (Take n = 40.) Round your answer to three decimal places.
__________
+129899 (a) Call the number who married in 1960 = M
And.....5 years later......the number who remained married = .928M
And we have that one year later 98% of these remain married....so....... 0.98 (.928M) = .90944M
So
.90944M = .928M*e^(k * 1) divide by M
.90944 = .928*e^(k) divide by .928
.90944/.928 = e^(k) take the ln of both sides
ln (.90944/.928) = ln e^(k)
k = ln (.90944/.928) = -.02 = the decay factor
(b) The function is .928M * e^(-.02n) where n is the time in years since the 5th anniversary
(c) 45 years later we have
.928M * e ^(-.02 * 40) ≈ .417M [about 41.% remain married ]
+129899 (a) Call the number who married in 1960 = M
And.....5 years later......the number who remained married = .928M
And we have that one year later 98% of these remain married....so....... 0.98 (.928M) = .90944M
So
.90944M = .928M*e^(k * 1) divide by M
.90944 = .928*e^(k) divide by .928
.90944/.928 = e^(k) take the ln of both sides
ln (.90944/.928) = ln e^(k)
k = ln (.90944/.928) = -.02 = the decay factor
(b) The function is .928M * e^(-.02n) where n is the time in years since the 5th anniversary
(c) 45 years later we have
.928M * e ^(-.02 * 40) ≈ .417M [about 41.% remain married ]
