Solve using the graphical method.

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

{“version”:”1.1″,”math”:”<math ><mi>a</mi><mo>.</mo><mo> </mo><mo> </mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>≥</mo><mn>140</mn><mspace linebreak=”newline”></mspace><mi>b</mi><mo>.</mo><mo> </mo><mo> </mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>≤</mo><mn>140</mn><mspace linebreak=”newline”></mspace><mi>c</mi><mo>.</mo><mo> </mo><mo> </mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>30</mn><mi>y</mi><mo>≥</mo><mn>180</mn><mspace linebreak=”newline”></mspace><mi>d</mi><mo>.</mo><mo> </mo><mo> </mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>30</mn><mi>y</mi><mo>≤</mo><mn>180</mn></math>”}

(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

{“version”:”1.1″,”math”:”<math ><mi>a</mi><mo>.</mo><mo> </mo><mo> </mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>≥</mo><mn>140</mn><mspace linebreak=”newline”></mspace><mi>b</mi><mo>.</mo><mo> </mo><mo> </mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mo>≤</mo><mn>140</mn><mspace linebreak=”newline”></mspace><mi>c</mi><mo>.</mo><mo> </mo><mo> </mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>30</mn><mi>y</mi><mo>≥</mo><mn>180</mn><mspace linebreak=”newline”></mspace><mi>d</mi><mo>.</mo><mo> </mo><mo> </mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>30</mn><mi>y</mi><mo>≤</mo><mn>180</mn></math>”}

(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

{“version”:”1.1″,”math”:”<math ><mi>a</mi><mo>.</mo><mo> </mo><mo> </mo><mi>C</mi><mo>=</mo><mn>10</mn><mi>x</mi><mo>+</mo><mn>30</mn><mi>y</mi><mspace linebreak=”newline”></mspace><mi>b</mi><mo>.</mo><mo> </mo><mo> </mo><mi>C</mi><mo>=</mo><mn>30</mn><mi>x</mi><mo>+</mo><mn>10</mn><mi>y</mi><mspace linebreak=”newline”></mspace><mi>c</mi><mo>.</mo><mo> </mo><mo> </mo><mi>C</mi><mo>=</mo><mn>500</mn><mi>x</mi><mo>+</mo><mn>300</mn><mi>y</mi><mspace linebreak=”newline”></mspace><mi>d</mi><mo>.</mo><mo> </mo><mo> </mo><mi>C</mi><mo>=</mo><mn>300</mn><mi>x</mi><mo>+</mo><mn>500</mn><mi>y</mi></math>”}

(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)

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