Jiffy-Pol Consultants is paid $1,000,000 for each percentage of the vote that Senator Sleaze receives in the upcoming election.Sleaze's share of the vote is determined by the number of slanderous campaign ads run by Jiffy-Pol according to the function S = 100N/(N + 1), where N is the number of ads.If each ad costs $4,900 approximately how many ads should Jiffy-Pol buy in order to maximize its profits?
A)2,853
B)1,428
C)98
D)1,477
E)714

Question

Quizplus · Accepted Answer

To maximize profits, we need to find the number of ads that will generate the highest payment from Senator Sleaze, while also minimizing the cost of purchasing the ads.

The payment from Senator Sleaze is given by the function S = $1,000,000 * F1/(N/F1 + 1), where F1 is the fraction of the vote that Sleaze receives and N is the number of ads.

We can rewrite this function as S = $1,000,000 * F1 - ($1,000,000 * F1/(N/F1 + 1)).

The cost of purchasing N ads is $4,900 * N.

So the total profit function for Jiffy-Pol is P = $1,000,000 * F1 - ($1,000,000 * F1/(N/F1 + 1)) - $4,900 * N.

To maximize profits, we need to find the number of ads that will make the derivative of P with respect to N equal to 0.

Taking the derivative of P with respect to N, we get:

P' = -($1,000,000 * F1 * (F1/(N/F1 + 1)^2))/F1^2 - $4,900

Setting P' equal to 0, we can solve for N:

-$1,000,000 * F1 * (F1/(N/F1 + 1)^2))/F1^2 - $4,900 = 0

Simplifying this equation, we get:

N = (1/F1 - 1) * $1,000,000/ $4,900

N = (1/F1 - 1) * 204.08

To maximize profits, we want to find the value of F1 that will make N as large as possible, while still being less than or equal to the total number of voters (because we can't buy more ads than there are voters).

The fraction of the vote that Sleaze will receive is maximized when F1 = 1, so we'll try that first:

N = (1/1 - 1) * 204.08 = undefined

This doesn't make sense - we can't have an undefined number of ads! So we need to try a smaller value of F1.

Let's try F1 = 0.9:

N = (1/0.9 - 1) * 204.08 = 28.53 * 204.08 = 5,827

This is more ads than there are voters, so we need to try a smaller value of F1.

Let's try F1 = 0.8:

N = (1/0.8 - 1) * 204.08 = 23.08 * 204.08 = 4,706

This is still more ads than there are voters, so we need to try a smaller value of F1.

Let's try F1 = 0.7:

N = (1/0.7 - 1) * 204.08 = 19.05 * 204.08 = 3,888

This is still more ads than there are voters, so we need to try a smaller value of F1.

Let's try F1 = 0.6:

N = (1/0.6 - 1) * 204.08 = 15.87 * 204.08 = 3,240

This is still more ads than there are voters, so we need to try a smaller value of F1.

Let's try F1 = 0.5:

N = (1/0.5 - 1) * 204.08 = 12.84 * 204.08 = 2,620

This is less than the total number of voters (which we don't know, but it's somewhere between 2,620 and 5,827). So we can purchase 2,620 ads and maximize our profits.

Therefore, the answer is B) 1,428 (rounded to the nearest whole number of ads).

Jiffy-Pol Consultants Is Paid $1,000,000 for Each Percentage of the Vote