6. The use of automatic teller machines is continuing a steady growth in the
Using data from the mid-1990’s (from the August 31, 1997, Cincinnati Enquirer),
a model tracking the annual number of transactions at ATMs nationwide (in
millions) is given by A(t) = 61.4t + 521.9 where t measures the number of years
since 1990. The same report yields the model formula p (t) = 0.0065t2 – 0.0553t +
0.682 for the percentage of ATM transactions in which customers withdrew cash
from their accounts.
(a) [5 points] Calculate A (5) and p(5) and interpret these numbers.
A(5) = 61.4 · 5 + 521.9 = 828.9 million transactions at ATMs in 1995
p(5) = 0.0065 · 52 – 0.0553 · 5 + 0.682 = .568 = 56.8% of ATM transactions were
with drawals in 1995
(b) [10 points] Let f(t) = A(t)·p(t) be the function obtained by multiplying the
functions A and p. What does the function f(t) measure? Calculate and
interpret the value of f (5).
f(t) = # of transactions at ATMs nationwide (in millions) which were cash
Therefore, f(5) = 470.8 million ATM transactions in 1995 were cash withdrawals.
2. [15 points] The practices of fishermen worldwide has
been a concern of
environmentalists for years. Data from the early 1990’s yields the following quadratic function as a model for the size of catches of fish (in millions of
tons) measured in year t after 1990: F(t) = 1.155t2 – 1.743t + 97.462. For
instance, in 1993, total world catches of fish amounted to F(3) = 102.6 million
metric tons. Find the vertex of theparabola which is the graph of the function
F(t) and explain what it tells us about world fishing.
The vertex has x-coordinate h = –b/2a = –(–1.743)/(2*1.155) = .75 and
= B(h) = 96.804. So the vertex is the point (.75, 96.8). This means that .75
1990 (by 1991) 96.8 million metric tons were fished worldwide, representing a
minimum value, and (since a = 1.155 is positive ) this total has been growing
6. [10 points] Census figures for the early 1990’s give data on the populations
each of the 50 states. The fol lowing exponential models give population
for the states of Ohio, Kentucky and Indiana (where t measures years since 1990,
and populations are measured in thousands):
(a) De termine the populations of each of the three states in 2001, based on
models. Also give the total population of the tristate area in 2001.
The population of Ohio in 2001 is = 11466.319 thousand = 11,466,319
The population of Kentucky in 2001 is = 4068.225 thousand = 4,068,225
The population of Indiana in 2001 is = 6106.421 thousand = 6,106,421
So the total population of the tristate is 21,640,965.
(b) Write a formula for the function P(t) which gives the total tristate
in year t. Calculate and interpret the quantity P(11).
From this we find that the total population of the Tristate is
P(11) = 21640.965 thousand = 21,640,965.
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