302 CHAPTER 3
■
Exponential Functions and Models
16. Half-Life of Radioactive Iodine A student is trying to determine the half-life of
radioactive iodine-131. She measures the amount of iodine-131 in a sample solution
every 8 hours. Her data are shown in the table.
(a) Make a scatter plot of the data.
(b) Use a calculator to find an exponential curve that models the amount
of iodine-131 remaining after x hours.
(c) Graph the model from part (b) together with the scatter plot in part (a). Use the
feature to determine the half-life of iodine-131.
TRACE
f 1x 2= b
#
a
x
17. Hybrid Car Sales The table shows the number of hybrid cars sold in the United
States for the period 2000–2007.
(a) Complete the table to find the one-year growth factor for each one-year time period.
For instance, the one-year growth factor for the time period 2000–2001 is the ratio of
the sales in 2001 to the sales in 2000, as calculated in the last two columns in the
table.
(b) Find the “average” one-year growth factor A. Do this by finding the average of all
the growth factors you found in the last column of the table.
(c) It seems reasonable to use the average growth factor in part (b) to construct an
exponential model of the form for the number of hybrid cars sold in
year x. Use your answer from part (b) to construct such a model.
(d) Use a graphing calculator to find the exponential model of best fit for the number
of hybrid cars sold in year x, where represents the year 2000.
(e) Graph the models from (c) and (d) together with a scatter plot of the sales data.
x = 0
f 1x 2= ba
x
Time (h) 0 8 16 24 32 40 48
Amount of (g)
131
I
4.80 4.66 4.51 4.39 4.29 4.14 4.04
Hybrid car sales
Year x
Number of hybrid cars
sold (thousands)
sales in year x ⴙ 1
sales in year x
Growth factor
2000 0 9.5
— —
2001 1 20.3
20.3>9.5
2.14
2002 2 35.0
2003 3 43.4
2004 4 85.0
2005 5 205.8
2006 6 254.5
2007 7 347.1
18. Population of Belgium Many highly developed countries, particularly in Europe, are
finding that their population growth rates are declining and that a logistic function
provides a much more accurate model than an exponential one for their population. The
table gives , the midyear population of Belgium (in millions), for years between
1980 and 2000, where t represents the number of years since 1980.
(a) Use a graphing calculator (TI-89 or better) to find a logistic model for Belgium’s
population.
B1t 2