3.1 Quadratic Functions Examples And Worksheet - Chapter 3: Polynomial And Rational Functions Page 7
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Chapter 3
Polynomial and Rational Functions
Example 6
Cost
A soft drink manufacturer has daily production costs of
2
C x
70,000
120x
0.055x
where C is the total cost (in dollars) and x is the number of units produced.
Estimate numerically the number of units that should be produced each day to
yield a minimum cost.
Solution
Figure 3.10
2
Enter the function
y
70,000
120x
0.055x
into your graphing utility. Then
use the table feature of the graphing utility to create a table. Set the table to start at
x
0
and set the table step to 100. By scrolling through the table you can see that
the minimum cost is between 1000 units and 1200 units, as shown in Figure 3.10.
You can improve this estimate by starting the table at
x
1000
and setting the table
step to 10. From the table in Figure 3.11, you can see that approximately 1090 units
should be produced to yield a minimum cost of $4545.50.
Now try Exercise 57.
Figure 3.11
Grants
Example 7
The numbers
g
of grants awarded from the National Endowment for the
Humanities fund from 1999 to 2003 can be approximated by the model
2
g t
99.14t
2,201.1t
10,896,
9 ≤ t ≤ 13
where represents the year, with
t
t
9
corresponding to 1999. Using this model,
determine the year in which the number of grants awarded was greatest.
(Source: U.S. National Endowment for the Arts)
Algebraic Solution
Graphical Solution
Use the fact that the maximum point of the
Use a graphing utility to graph
parabola occurs when
t
b 2a .
For this
2
y
99.14x
2,201.1x
10,896
function,
you
have
a
99.14
and
b
2201.1.
So,
for
9 ≤ x ≤ 13,
as shown in Figure 3.12. Use the maximum feature
(see Figure 3.12) or the zoom and trace features (see Figure 3.13) of the
b
t
graphing utility to approximate the maximum point of the parabola to
2a
be
x
11.1.
So, you can conclude that the greatest number of grants
were awarded during 2001.
2201.1
2
99.14
2000
1321.171
y = −99.14x
+ 2,201.1x − 10,896
2
11.1
From this -value and the fact that
t
t
9
represents 1999, you can conclude that the
greatest number of grants were awarded
9
13
11.0999
11.1020
during 2001.
0
1321.170
Figure 3.12
Figure 3.13
Now try Exercise 61.
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