Problem 2<=
/b>
A=
store
owner plans to sell two types of MP3 players. One type costs the store owner $200
each; the other type costs $400 each.
The $200 models yield a profit of $25 dollars each and the $400 mode=
ls a
profit of $30 each. The store=
owner
estimates that the total demand for the MP3 players will not exceed 300
units. The store owner can in=
vest
up to $80,000 on the inventory of MP3 players. How many of each model should the =
store
owner sell to maximize profit and what is the maximum profit?
<=
o:p>
[Problem submitted=
by Vin
Lee, LACC Associate Professor of Mathematics.]
Solution for Prob=
lem 2:
Thi=
s is a
linear programming problem. L=
et x
be the number of MP3 players that cost $200 each, y the number of MP3 playe=
rs
that cost $400 each, and z the profit.&nbs=
p;
The profit is![](3D"2007prob2_files/image002.gif")
. The const=
raints
are:<=
o:p>
1)<=
span
style=3D'mso-spacerun:yes'> ![](3D"2007prob2_files/image004.gif")
=
2)<=
span
style=3D'mso-spacerun:yes'> ![](3D"2007prob2_files/image006.gif")
3)<=
span
style=3D'mso-spacerun:yes'> ![](3D"2007prob2_files/image008.gif")
=
![](3D"2007prob2_files/image010.gif")
=
![](3D"2007prob2_files/image012.gif")
4)<=
span
style=3D'mso-spacerun:yes'> ![](3D"2007prob2_files/image014.gif")
=
![](3D"2007prob2_files/image017.gif")
The=
number
of each type of MP3 player that will maximize the profit will be the
coordinates of one of the vertices of the region satisfying all four of the
constraints.
![](3D"2007prob2_files/image019.jpg")
If
(x,y) =3D (0,0), then &=
nbsp; &nbs=
p; ![](3D"2007prob2_files/image021.gif")
.
If (x,y) =3D (0,200), then ![](3D"2007prob2_files/image023.gif")
.
If (x,y) =3D (200,100), then =
![](3D"2007prob2_files/image025.gif")
.
If (x,y) =3D (300,0), then ![](3D"2007prob2_files/image027.gif")
.
So,=
the
maximum profit is $8000 which is attained when 200 of the $200 MP3 players =
and
100 of the $400 MP3 players are sold.
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