A math 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 monthly salary of an aide is $1100. The camp can accommodate up to 45 staff members and needs 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? a. Define variables. b. Write a system of linear inequalities to model this
problem. c. Graph the system and find the corner points. d. Find the solution and state using a complete sentence.
(a) Let x counselors and y aides be hired
(b) Objective Function:
Minimize C = 2400x + 1100y
Constraint Inequalities:
30 ≤ (x + y) ≤ 45 [Accommodation] (This can be split as x + y ≤ 45 and x + y  30)
y  10 [Minimum number of Aides]
y/x ≤ 3/2, that is 3x - 2y  0 [Counselors to Aides ratio]
x, y  0 [Non-negativity]
(c) Graph:
The corner points are in the table below:
(d) Optimal solution: x = 12, y = 18, C = $48600
The camp should hire 12 Counselors and 18 Aides. The minimum cost will be $48600.