Two candidates are running for mayor in a small town. The campaign committee for candidate A has been conducting weekly telephone polls to assess the progress of the campaign. Currently, there are 16,000 registered voters, 43% of whom are planning to vote. Of those planning to vote, 59% will vote for candidate A. Candidate B has begun some serious mudslinging, which has resulted in increasing public interest in the election and decreasing support for candidate A. Polls show that the percentage of people who plan to vote is increasing by 5 percentage points per week, and the percentage who will vote for candidate A is declining by 4 percentage points per week. How rapidly is the number of votes that candidate A will receive increasing at this moment? (Answer in the nearest integer.)

a) 6,880

b) 4,059

c) Check Explanation

The number of expected votes for candidate A increases only in the first 3 weeks of mudslinging. The rate of weekly increase in those 3 weeks, is provided in the explanation. The number changes weekly for those 3 weeks with an average increase of 101 new votes per week.

Step-by-step explanation:

a. If the election were held today, how many people would vote?

b. How many of those would vote for candidate A?

c. How rapidly is the number of votes that candidate A will receive increasing at the moment?

There are 16,000 registered voters, 43% of whom are planning to vote, with 59% planning to vote for candidate A.

a) Number of registered voters planning to vote = 43% × 16000 = 6880

b) Number of registered voters that will vote and vote for candidate A

= 59% of registered voters planning to vote

= 59% × 6880 = 4059.2 ≈ 4059 people

c) Polls show that the percentage of people who plan to vote is increasing by 5 percentage points per week, and the percentage who will vote for candidate A is declining by 4 percentage points per week.

Since, the 'moment' isn't specified, we will check how much the number is increasing for the first 4 weeks after the mudslinging by candidate B began

Normally, 43% of registered voters want to vote, but now it is increasing at a rate of 5% per week. So, the percentage of registered voters that want to vote is now

43% + 5x% (where x = number of weeks after the mudslinging by candidate B started)

And the percentage of voting, registered voters that want to vote for candidate A is now (59% - 4x%)

After a week, percentage of registered voters that will vote = 48%

Number of registered voters that will vote = 48% × 16000 = 7680

percentage of voting, registered voters that want to vote for candidate A = 55%

Number of voting, registered voters that want to vote for candidate A = 55% × 7680 = 4224

Difference between the initial number of expected votes for candidate A between the beginning of the mudslinging and end of week 1

= 4224 - 4059 = 165

After week 2,

percentage of registered voters that will vote = 53%

Number of registered voters that will vote = 53% × 16000 = 8480

percentage of voting, registered voters that want to vote for candidate A = 51%

Number of voting, registered voters that want to vote for candidate A = 51% × 8480 = 4324.8 = 4325

Difference between the number of expected votes for candidate A between week 1 and week 2

= 4325 - 4224 = 101

After week 3,

percentage of registered voters that will vote = 58%

Number of registered voters that will vote = 58% × 16000 = 9280

percentage of voting, registered voters that want to vote for candidate A = 47%

Number of voting, registered voters that want to vote for candidate A = 47% × 9280 = 4361.6 = 4362

Difference between the number of expected votes for candidate A between week 2 and week 3

= 4362 - 4325 = 37

After week 4,

percentage of registered voters that will vote = 63%

Number of registered voters that will vote = 63% × 16000 = 10,080

percentage of voting, registered voters that want to vote for candidate A = 43%

Number of voting, registered voters that want to vote for candidate A = 43% × 10080 = 4334

Difference between the number of expected votes for candidate A between week 3 and week 4

= 4334 - 4362 = -28

The number of expected votes for candidate A begins to decline after the 4th week of mudslinging.

So, the required 'moment' should be within the first 3 weeks of mudslinging. And the rate of increase weekly is provided above with an average increase of 101 new voters per week.

Hope this Helps!!!

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