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 $2,000 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; 4,000 flyers should be mailed. B) 100 hours of speeches should be given; 2,000 flyers should be mailed out. C) 50 hours of speeches should be given; 3,000 flyers should be mailed out. D) 500 hours of speeches should be given; 1,500 flyers should be mailed out. E) 200 hours speeches should be given; no flyers should be mailed out.

Question

Quizplus · Accepted Answer

To maximize votes, we need to maximize the value of V. Let's calculate the cost of making speeches and mailing flyers: - Cost of making speeches: 10S - Cost of mailing flyers: 0.5M The total cost of the campaign cannot exceed $2,000, so we have the inequality: 10S + 0.5M ≤ 2,000 Solving for M, we get: M ≤ 4,000 - 20S Now let's substitute this expression for M in the equation defining V: V = 500S^0.20M^0.60 V = 500S((S/100) + 1)P10.20(S/100)P0.60(4,000 - 20S)/100 Simplifying this expression, we get: V = 50(S^2 - 20S + 80,000) Now we can take the derivative of V with respect to S, set it equal to zero to find the critical point, and check whether it's a maximum. We get: dV/dS = 100S - 2,000 = 0 S = 20 We can check that this is a maximum by computing the second derivative of V with respect to S: d^2V/dS^2 = 100 > 0 Since the second derivative is positive, the critical point S = 20 is a maximum of V. Now we can substitute S = 20 in the expression we found for M: M ≤ 4,000 - 20(20) = 3,600 To maximize V, we should choose the highest possible value of M that satisfies this inequality, which is M = 3,000. Therefore, the best allocation of the budget is: - 50 hours of making speeches - 3,000 flyers mailed This gives a value of V = 522,000.

A Politician Facing Reelection Can Win Votes According to the Following