Math 093 - Exponential functions
Name___________________________________
Solve the problem.
1) A small music store's annual sales were $289 thousand in 2003 and have grown by about 19% per year since
then.
a) Let F(x) represent the annual sales in thousand dollars at x years since 2003. Construct the exponential
function F(x) = a·bx that models the situation at x years since 2003.
b) Predict when the annual sales will reach $1 million. (1 million = 1000 thousands)
c) Another possible equation for this situation is f(x) = 289e0.174x. Find f(10) and interpret in context.
2) Revenue from board games has decayed approximately exponentially from $12 million in 1990 to $6 million in
2007.
a) Use algebra to construct the exponential model for the data at x years since 1990.
b) Predict when the revenue will reach $4 million.
c) Another model for the data is f(x) = 12e-0.041x . Find f(10) and interpret in context using correct units.
d) Use the function from part (c) to solve f(t) = 9. Interpret your result in context using correct units.
3) The value of a particular investment follows a pattern of exponential growth. In the year 2000, you invested
money in a money market account. The value of your investment, in dollars, at t years after 2000 is given by the
exponential growth model f(t) = 3900e0.049t.
(a) How much did you initially invest in the account?
(b) When will the amount in the account double?
(c) Interpret in context the meaning of the mathematical statement f(5) = 4982.72.
(d) Another possible model for the situation is f(t) = 3900*(1.05)^t. Use this function to find f(8). Interpret in
context.
(e) Use the model from part (d) to find t when f(t) = 6050.18. Interpret in context.
4) Since the late 1990's electronic payments made to an online company have increased exponentially.
Year Number of Electronic Payments
(thousands)
1999
53
2001
97
2003
185
2005
354
2006
487
a) Let f(t) be the number of electronic payments (in thousands) in the year that is t years since 1995. Use the
regression feature of your calculator to produce the best model that fits the data.
b) Another possible equation of f is f(t) = 14.38e0.32t. Predict in which year there will be 4 million (4000
thousands) electronic payments.
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5) Since the late 1990's electronic payments made to an online company have increased exponentially.
Year Number of Electronic Payments
(thousands)
1999
53
2001
97
2003
185
2005
354
2006
487
a) Let f(t) represent the number of electronic payments (in thousands) at t years since 1999. Use the calculator to
construct a scatter-diagram for the data. Show rough sketch, label axes.
b) What is the best model to fit the data: linear or exponential?
c) Use the regression feature of the calculator to find the best model that fits the data; round to three decimal
places.
d) What is the coefficient? What does it mean in the situation?
e) What is the base of the exponential function? Interpret as a multiplier. Interpret as a rate.
f) Use the first and last points on the table and ALGEBRA to construct the exponential model that fits the data at
t years since 1999.
g) Let f(t) be the number of electronic payments in the year that is t years since 1999. Another possible equation
of f is f(t) = 52.215e0.318x. Predict in which year there will be 913 thousand elctronic payments.
6) The function y = 300e-0.0099x models the amount in pounds of a particular radioactive material stored in a
concrete vault, where x is the number of years since the material was put into the vault. How many pounds will
be left after 90 years?
7) General Insurance offers a $500,000 life insurance policy. Monthly rates for women and men are shown in the
table for variouus ages.
Age
30
35
40
45
50
55
60
65
Monthly Rate (dollars)
Women
Men
7.25
8.00
8.25
9.75
11.25
12.00
17.00
19.25
23.75
29.25
35.00
48.50
49.25
77.00
73.50
118.50
Let W(t) and M(t) be the monthly rates (in dollars) for women and men, respectively, both at t years of age.
a) Find the equation for W.
b) Predict how much a 52-year-old woman would pay per month for a $500,000 policy.
c) Predict the age of a woman who pays $50 per month for a $500,000 policy.
d) Find the regression equation for M.
e) Predict how much a 37-year-old man would pay per month for a $500,000 policy.
f) Predict the age of a man who pays $50 per month for a $500,000 policy.
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8) The number of cases of a certain strain of influenza in the world are shown in the table for various years.
Year Number of Cases (thousands)
1993
283
1995
137
1998
63
2001
27
2004
19
Let f(t) be the number of cases (in thousands) in the year that is t years from 1990.
a) Use a graphing calculator to find the equation of f.
b) Predict how many cases there will be (to the nearest thousand) in 2006.
c) Predict when there will be 42 thousand cases in the world.
9) SOME OF THE ANSWERS - If there are mistakes, let me know.
(1)F(t) = 289*(1.19)^t; (b) sometime in the year 2010 ;
thousands dollars (1.646 million dollars)
(c) (10, 1646.532). In 2013 the annual sales reached 1646
(2) (a) y = 12 (0.96)^x; (b) 2017; (c) f(10) = 7.98. In the year 2000 the revenue from board games was 7.98
million dollars. million; (d) t = 7. In the year 1997 the revenue from board games was 9 million dollars.
(3) a) $3900; (b) sometime in 2014; (c) In 2005 there were $4982.72 in the account; (d) $5762.08. In 2008 there
were $5762.08 in the account; (e) t = 9. In the year 2009 the account had $6050.18
(4) (b) 2012
(5) b) exponential; (c) f(t) = 52.215(1.375)^t; (d) a = 52.215. (0, 52.215). In the year 1999 there were 52,215
electronic payments made to an online (e) b = 1.375. Every year since 1999 the number of electronic payments
has been increasing by multiplying by 1.375; which represents an annual rate of 37.5%. (f) f(t) = 53*(1.373)^t,
(g) In the year 2008.
(6) 123 pounds
(7) (a-b) W(t) = 0.79(1.07)t; $26.64; (d-e) M(t) = 0.58(1.08)t; 10.00;
(8) (a) f(t) = 508.6368(0.77956)t; (b) 9,500 cases; (c) year 2000
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