Give the mathematical formulation of the linear programming problem. Graph the f

Question

Give the mathematical formulation of the linear programming problem. Graph the feasible region described by the constraints and find the corner points. Do not attempt to solve:

1a) A summer camp wants to hire counselors and aides to fill its staffing needs at minimum cost. The average monthly salary of a counselor is $2400 and the average aide salary is $1100. the camp can accomommodate up to 45 staff memebers and need at least 30 to run properly. They must have at least 10 aides, and may have up to 3 aides for every 2 counselors. How many counselors and how many aides should the camp hire to minimize cost?

1b)Use graphical methods to solve the linear programming problem.

Minimize subject to:
z=4x+5y

Explanation / Answer

you want the minimum number of employees, which is 30 (the minimum number of employees also means the minimum number of counselors, so have 3/2 as many aides [or fewer] to minimize the number of counselors). So see if you can solve for 30. let x=number of counselors. then y= number of aides= (3/2)x x + (3/2)x= (5/2)x = 30 Then x=12 and y=(3/2)12= 18. So you see that the number of employees must have been a multiple of 5, because 5 must have divided the number of employees in the last equation. If it was 5 aides for every 3 counselors, then x + (5/3)x= (8/3)x which would not divide 30. Then in this case x= 12, and y does not equal 20 to minimize. It would obviously equal 18, since you can get away with fewer aides. So in general, to minimize let x= number of counselors, y= number of aides= (ratio)x, where ratio is the ratio of aides to counselors. If you let the ratio be equal to ratio', where ratio'

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