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

To maximize votes, we need to find the budget allocation between speeches and mailing flyers that results in the highest total vote count. Let V(S,M) denote the vote count function. Then, V(S,M) = 500S*S1*U1*P10.20*S1*S1*P0M*S1*U1*P10.60*S1*S1*P0 where S1 = 1 if S>0 and S1=0 otherwise, and similarly for each of the other indicator variables. To maximize V(S,M) subject to the budget constraint, we need to use Lagrange multipliers: L(S,M,λ) = V(S,M) + λ(8000 - 10S - 0.5M) Taking partial derivatives, ∂L/∂S = 1000S^0.20 - 10λ = 0 ∂L/∂M = 2500S^0.60 - 0.5λ = 0 ∂L/∂λ = 8000 - 10S - 0.5M = 0 Solving these equations, we get: S = 200 hours M = 12000 flyers Therefore, choice C is the best one.

A Politician Facing Reelection Can Win Votes According to the Following