Question

1. A company produces two types of nutritional supplements; Energize and Excel. Energize contains 26 mg of vitamin A, 42 mg of vitamin C and 25 mg of an herbal supplement. Excel contains 68 mg of vitamin A, 61 mg of vitamin C and 15 mg of the herbal supplement. An athlete is told that he needs at least 489 mg of vitamin A, 1130 mg of vitamin C and 348 mg of the herbal supplement for optimal athletic performance. The athlete wants to take the supplements, but at the lowest possible cost. Energize pills cost 40 cents each, while Excel pills cost 100 cents each. Let x = the number of Energize pills to take, and let y = the number of Excel pills to take. Which of the following is the objective function for this situation?
A) Minimize C = 40x + 100y.
B) Maximize C = 40x + 100y.
C) Minimize C = 11300x + 3480y.
D) Maximize C = 100x + 40y.
E) Minimize C = 100x + 40y.
F) None of the above.
2. A company produces two types of bicycles; mountain bikes and racing bikes. It takes 7 hours of assembly time and 5 hours of mechanical tuning to produce a mountain bike. It takes 8 hours of assembly time and 4 hours of mechanical tuning to produce a racing bike. The company has at most 22 hours of mechanical tuning labor per week and at most 190 hours of assembly labor per week. The company's profit is $50 for each mountain bike produced and $130 for each racing bike produced. The company wants to make as much money as possible. Let x = the number of mountain bikes they produce, and let y = the number of racing bikes they produce. What are the constraints for this problem?
a) 7x + 5y ≤ 190, 8x + 4y ≤ 22, x ≥ 0, y ≥ 0.
b) 7x + 8y ≥ 190, 5x + 4y ≤ 22, x ≥ 0, y ≥ 0.
c) 5x + 8y ≤ 190, 7x + 4y ≤ 22, x ≥ 0, y ≥ 0.
d) 7x + 8y ≤ 190, 5x + 4y ≤ 22, x ≥ 0, y ≥ 0.
e) 7x + 8y ≤ 190, 5x + 4y ≥ 22, x ≥ 0, y ≥ 0.
f) none of the above.
3. A certain academic department at a local university will conduct a research project. The department will need to hire graduate research assistants and professional researchers. Each graduate research assistant will need to work 23 hours per week on fieldwork and 17 hours per week at the university's research center. Each professional researcher will need to work 15 hours per week on fieldwork and 25 hours per week at the university's research center. The minimum number of hours needed per week for fieldwork is 154 and the minimum number of hours needed per week at the research center is 140. Each research assistant will be paid $295 per week and each professional researcher will be paid $416 per week. Let xdenote the number of graduate research assistants hired and let y denote the number of professional researcher hired. The department wants to minimize cost. Set up the Linear Programming Problem for this situation.
A) Min C = 416x + 295y; s.t 23x + 17y ≥ 154, 15x + 25y ≥ 140, x ≥ 0, y ≥ 0.
B) Min C = 295x + 416y; s.t 23x + 15y ≥ 154, 17x + 25y ≥ 140, x ≥ 0, y ≥ 0.
C) Min C = 295x + 416y; s.t 23x + 15y ≥ 154, 25x + 17y ≥ 140, x ≥ 0, y ≥ 0.
D) Min C = 295x + 416y; s.t 15x + 23y ≤ 140, 25x + 17y ≤ 154, x ≥ 0, y ≥ 0.
E) Min C = 416x + 295y; s.t 15x + 23y ≥ 154, 25x + 17y ≥ 140, x ≥ 0, y ≥ 0.
F) None of the above.

Answer 1

Answer:

1)A) Minimize C = 40x + 100y.

2) d) 7x + 8y ≤ 190, 5x + 4y ≤ 22, x ≥ 0, y ≥ 0.

3) B) Min C = 295x + 416y; s.t 23x + 15y ≥ 154, 17x + 25y ≥ 140, x ≥ 0, y ≥ 0.

Step-by-step explanation:

1) Energize pills cost 40 cents each, while Excel pills cost 100 cents each. If x represent the energize pills and y represent the excel pills, the objective function would be:

minimize cost = cost of energize pill + cost of excel pill

Minimize cost = 40x + 100y

2) Let x = the number of mountain bikes they produce, and let y = the number of racing bikes they produce.

7 hours of assemble time to produce a mountain bike and 8 hours of assemble time to produce a racing bike. Since the company has at most 190 hours for assemble time, it can be represented as:

7x + 8y≤ 190

5 hours of mechanical tuning to produce a mountain bike and 4 hours of mechanical tuning to produce a racing bike. Since the company has at most 22 hours for mechanical tuning, it can be represented as:

5x + 4y ≤ 22

Also assemble time and mechanical time must be greater than 0, hence:

x ≥ 0, y ≥ 0

3) The fieldwork constraint can be represented as:

23x + 15y ≥ 154

The time to be spent at university research center can be represented as:

17x + 25y ≥ 140

Both fieldwork time and time spent at university research center mut be greater than 0, hence:

x ≥ 0, y≥ 0

The cost equation is:

Minimize cost = 295x + 416y