Transcript Constructing Functions

Sullivan PreCalculus
Section 2.6
Mathematical Models:
Constructing Functions
Objectives
• Construct and analyze functions
Example: The price p and the quantity x sold of a
certain product obey the demand equation
1
p   x  100
4
0  x  400
a.) Express the revenue R as a function of the
quantity of items sold (x).
Revenue = (Price)(Items sold) = (x)(p)
1

R( x )  x  x  100
 4

1 2
  x  100 x
4
b.) What is the revenue if 150 items are sold?
1
R(150)   (150) 2  100(150)  $9,375
4
c.) Graph the function R(x) on a graphing utility.
d.) Using the graph, find the number of items x that
will maximize revenue. What is the maximum
revenue?
Quantity that maximizes revenue: 200 items
Maximum Revenue: $10,000
e. What price should be charged for each item to
achieve maximum revenue?
Maximum Revenue occurs when x = 200 items
1
Price = p   x  100 0  x  400
4
1
p   (200)  100  $50
4
$50 should be charged to achieve maximum revenue.
Example: An open box with a square base is to
made from a square piece of cardboard 30 inches on
a side by cutting out a square from each corner and
turning up the sides.
a.) Express the volume V of the box as a
function of the length x of the side of the
square cut from each corner.
The volume of a box is given by:
V = (length)(width)(height)
x
x
x
x
30 in.
x
x
x
Length = 30 - 2x
30in.
x
Width = 30 - 2x
Height = x
So, Volume = (30 - 2x)(30 - 2x)(x)
c.) Graph V(x) using a graphing utility and
estimate what value of x will maximize V.
At x = 5 inches, the volume is maximum
(2000 cubic inches)
Example
A rectangle has one corner on the graph of y  16  x 2,
another at the origin, and third on the positive y axis and
fourth on the positive x axis.
(0,16)
(x,y)
y  16  x 2
a. Express the area of the rectangle as a function of x.
A = lw
A = xy
A = x(16  x 2 )
since
y  16  x 2
b. Find the domain of the function.
Since Area must be positive, then x and y are positive.
y  16  x 2  0
x 2  16
0 x4
or
(0,4)