A politician facing reelection can win votes according to the following process: V = 500S^0.20M^0.60, where S is hours of making campaign speeches and M is the number of flyers mailed.Making speeches costs $10 per hour, mailing flyers costs $.50 per flyer, and $8,000 are available to spend on the campaign.Assuming the politician wants to maximize votes, how should the budget be allocated between speeches and mailing flyers? A)No speeches should be given; 16,000 flyers should be mailed. B)400 hours of speeches should be given; 8,000 flyers should be mailed out. C)200 hours of speeches should be given; 12,000 flyers should be mailed out. D)2,000 hours of speeches should be given; 6,000 flyers should be mailed out. E)800 hours speeches should be given; no flyers should be mailed out.

Question

Quizplus · Accepted Answer

The objective of the politician is to maximize the number of votes. The formula indicates how many votes are obtained for a certain number of hours of speeches and a certain number of flyers mailed out. The budget for the campaign is $8,000. Let x be the number of hours spent on speeches and y be the number of flyers mailed out. The cost of speeches is $10 per hour and the cost of mailing flyers is $.50 per flyer. Thus, the total cost of the campaign is: Cost = 10x + 0.5y Since the budget is $8,000, we have: 10x + 0.5y ≤ 8000 Solving for y, we get: y ≤ 16000 - 20x The formula for the number of votes is: V = F1F1F1S1F1F1F10 500S^0.20M^0.60 Substituting y for the maximum number of flyers that can be mailed out, we get: V = F1F1F1S1F1F1F10 500(16000 - 20x)^0.20M^0.60 Simplifying, we get: V = 8000S - 0.01x^2 + 16000x The objective is to maximize V. To find the values of x and y that maximize V, we need to take the derivative of V with respect to x and set it equal to zero: dV/dx = 16000 - 0.02x = 0 Solving for x, we get: x = 80000/2 = 4000 Substituting x = 4000 into the inequality for y, we get: y ≤ 16000 - 20(4000) = 0 Since y cannot be negative, we have: y = 12000 Therefore, the budget should be allocated as follows: 200 hours on speeches ($2,000) 12,000 flyers mailed out ($6,000) This allocation gives the maximum number of votes. Thus, the answer is C.

A Politician Facing Reelection Can Win Votes According to the Following