Section 3.6 (Sullivan, 8
th
ed.)
MAC1140, MAC1147
36:
Mathematical Models: Constructing Functions
For any quadratic equation f(x) = ax
2
+ bx + c, it will have its minimum or maximum at
its vertex.
It will be a maximum if a < 0 because the parabola will open downward; it
will be a minimum if a > 0.
The vertex can be found without graphing:
Vertex =


F
H
G
I
K
J
F
H
G
I
K
J
b
a
f
b
a
2
2
,
.
Axis of symmetry: the line x =

b
a
2
.
1.
Volume of a Cylinder:
If the volume of a cylinder is
V =
π
r
2
h
and the height is half
the radius, express the volume V as a function of
r
.
2.
Finding the area of a rectangular field with fixed perimeter:
George has 2000 feet of fencing available to enclose a rectangular field.
One side of
the field lies along a river, so only three sides require fencing.
a)
Express the area A of the rectangle as a function of x, where x is the length:
b)
What is the largest area?
3.
A rectangle has one corner on the graph of
y = 36  x
2
, x > 0, at (x, y); another at the
origin; and a third on the positive xaxis at (x, 0), and a fourth on the positive yaxis.
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 Fall '08
 Staff
 Math, Rectangle, rectangular field

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