linear programming word problem

[Guest]

Hello, I am having trouble determing what to set as x and y variables. i have tried setting amount in thousands deposited in Wachovia as X and amount in thousands deposited in CCB as Y. The equations that I keep returning to are 14x+8y=22000 and 7x+15y=22000. I know these are wrong but I am stuck. Please Help?!

Problem:

Mr. Jones wants to invest up to $22000 in two accounts. He will invest at least $2000 and up to $14000 in Wachovia, and he will invest up to $15000 in CCB. What amounts will yield the greatest income(interest). What is the maximum interest.

I need to determine two equations so that I can show a graph and solve the problem. I appreciate any help you can give me.

 

[pka]

First of all, you have not given us a complete problem.
What you have given us is:
\(\displaystyle \begin{array}{l}
x + y = 22000 \\
2000 \le x \le 14000 \\
0 \le y \le 15000 \\
\end{array}\)

 

[skeeter]

First of all, you have not given us a complete problem.

to elaborate on pka's statement ... the interest rates from each institution would help a great deal in solving this problem.

 

[soroban]

Hello, Brandon!

You left out some information . . .

Problem: Mr. Jones wants to invest up to $22000 in two accounts.
He will invest at least $2000 and up to $14000 in Wachovia,
and he will invest up to $15000 in CCB.
What amounts will yield the greatest income (interest)?
What is the maximum interest?

I've set \(\displaystyle x\) = amount (in $1000s) invested in Wachovia and \(\displaystyle y\) = amount (in $1000s) in CCB.

The equations that I keep returning to are \(\displaystyle 14x\,+\,8y\:=\:22000\) and \(\displaystyle 7x\,+\,15y\:=\:22000\:\) ??
What are the 14, 8, 7, and 15?
I know these are wrong . . . Yes, they are.

You forgot to give us the interest rates for the two companies.
No matter, we can still set up the problem . . .

Since he won't invest a negative amount of money: \(\displaystyle \,x\,\geq\,0,\;y\,\geq\,0\)
\(\displaystyle \;\;\)The graph is in the first quadrant.

"Mr. Jones wants to invest up to $22000 in two accounts."
\(\displaystyle \;\;\)This means: \(\displaystyle \,x\,+\,y\:\leq\:22\)
Graph the line: \(\displaystyle \,x\,+\,y\:=\:22\)
\(\displaystyle \;\;\\)It has intercepts \(\displaystyle (22,0)]\) and \(\displaystyle (0,22)\).
\(\displaystyle \;\;\)Shade the region below the line.

"He will invest at least $2000 and up to $14000 in Wachovia."
\(\displaystyle \;\;\)This means: \(\displaystyle \,2\:\leq\:x\:\leq\:14\)
Graph the two vertical lines: \(\displaystyle x\,=\,2\) and \(\displaystyle x\,=\,14\).
\(\displaystyle \;\;\)Shade the region between the two vertical lines.

"and he will invest up to $15000 in CCB."
\(\displaystyle \;\;\)This means: \(\displaystyle \,y \:\leq\:15\)
Graph the horizontal line: \(\displaystyle y\,=\,15\).
\(\displaystyle \;\;\)Shade the region below this line.

        |
      22*
        | \ |             |
        |   \             |
        |   | \           |
        |   |   \         |
        |   |     \       | 
      15+ - o - - - o - - + - -
        |   |:::::::::\   |
        |   |:::::::::::\ |
        |   |:::::::::::::o
        |   |:::::::::::::| \
        |   |:::::::::::::|   \
        |   |:::::::::::::|     \
      - + - o - - - - - - o - - - * - -
        |   2            14      22

The final shaded region represents all the points that satisfy all the inequalities.
But we are interested in only the vertices of this region.

Using some algebra (intersections of lines), we determine the vertices.
Moving counterclockwise from the lower left corner,
\(\displaystyle \;\;\)the vertices are: \(\displaystyle \,(2,0),\;(14,0),\;(14,8),\;(7,15),\;(2,15)\)

Test them in the Interest Function (which you know)
\(\displaystyle \;\;\) and see which one produces maximum interest.