Question

Solve using the graphical method. Choose your variables, identify the objective function and the constraints, graph the constraints, shade the feasibility region, identify all corner points, and determine the solution that optimizes the objective function. Use this information to answer the following 8-part question:

A city council voted to conduct a study on inner-city community problems using sociologists and research assistants from UMUC. Allocation of time and costs per week are given in the following table:

	Labor hours - Sociologist	Labor hours – research assistant	MINIMUM labor hours needed per week
Fieldwork	10	30	180
Research center	30	10	140
Costs per week	$500	$300

(i) Which variable should x define?(Enter letter for correct answer in the blank)

a. total amount of labor hours needed for sociologists

b. number of sociologists needed for the study

c. total number of labor-hours needed for research assistants

d. total number of study team members needed

(ii) Which variable should y define?(Enter letter for correct answer in the blank)

a. total number of study team members needed

b. total number of labor-hours needed for research assistants

c. total number of labor-hours needed for sociologists

d. number of research assistants needed for the study

(iii) Which of the following correctly defines the constraint on number of labor hours for fieldwork?(Enter letter for correct answer in the blank)

a.  30x+10y≥140 b.  30x+10y≤140 c.  10x+30y≥180 d.  10x+30y≤180

(iv) Which of the following correctly defines the constraint on labor-hours needed at the research center?(Enter letter for correct answer in the blank)

a.  30x+10y≥140 b.  30x+10y≤140 c.  10x+30y≥180 d.  10x+30y≤180

(v) Which of the following correctly defines the objective function? (Enter letter for correct answer in the blank)

a.  C=10x+30y b.  C=30x+10y c.  C=500x+300y d.  C=300x+500y

(vi) Which of the following sets of corner points correctly defines the feasible region for this system of linear inequalities? (Enter letter for correct answer in the blank)

a.  (0, 14) ; (0, 6) ; (3, 5)b.  (0, 14) ; (3, 5) ; (18, 0)c.  (0, 6) ; (3, 5) ; (18, 0)d.  (0, 6) ; (3, 5) ; (5, 0) ; (0, 0)

(vii) Based on your determination of the feasible region for this business process, how many sociologists and how many research assistants should be hired to complete this study for minimum cost? (Enter letter for correct answer in the blank)

a. 3 sociologists and 5 research assistants

b. 0 sociologists and 14 research assistants

c. 18 sociologists and 0 research assistants

d. None of these answers are correct

(viii) Which of the following is the correct value for the minimum cost to complete this study?

a.  C=$2500b.  C=$3000c.  C=$4200d.  C=$9000

Answer 1

(i) a) Total number of labor hours needed for sociologists.

(ii) b) Total number of labor hours needed for research assistants.

(iii) c) The constraint on the number of labor hours for fieldwork-

(iv) a) The constraint on the number of labor hours for research center -

  

(v) Objective function is -

c)

The graph is given below -

(vi) From the graph, corner points are given below -

b)   

(vii) For point (0,14)

For point (3,5)

For point (18,0)

So cost is minimum at point (3,5)

a. 3 sociologists and 5 research assistants

(viii) b) C = $ 3000