Problem 77 - Precalculus

Question

Registered Voters. According to the U.S. Census Bureau (www.ccnsus.gov), in the 2000 national election, \(58.9 \%\) of men over the age of 18 were registered voters, but only \(41.4 \%\) voted; and \(62.8 \%\) of women over 18 were registered voters, but only \(43 \%\) actually voted. Write a \(2 \times 2\) matrix with the following data: \(A=\left[\begin{array}{cc}\text { Percentage of registered } & \text { Percentage of registered } \\ \text { male voters } & \text { female voters } \\\ \text { Percent of males } & \text { Percent of femalcs } \\ \text { who voted } & \text { who voted }\end{array}\right]\) If we let \(B\) be a \(2 \times 1\) matrix representing the total population of males and females over the age of 18 in the United States, or \(B=\left[\begin{array}{l}100 \mathrm{M} \\ 110 \mathrm{M}\end{array}\right],\) what docs \(A B\) tell us?

Step-by-Step Solution

Answer

Matrix \(A B\) gives the number of registered and actual voters in millions among men and women.

Step 1: Setup Matrix A

To create matrix A, we need to organize the information into a 2x2 format as follows:\[A = \begin{bmatrix}58.9 & 62.8 \41.4 & 43\end{bmatrix}\]- First row: percentages of registered male and female voters.- Second row: percentages of male and female voters who actually voted.

Step 2: Setup Matrix B

Matrix B represents the total population of males and females over 18 in millions. It is structured as follows:\[B = \begin{bmatrix}100 \ 110\end{bmatrix}\]Here, 100M refers to males and 110M refers to females.

Step 3: Multiply Matrices A and B

To find \(A \times B\), we multiply each element of row of A by the corresponding element of column of B and sum them up:\[AB = \begin{bmatrix}58.9 & 62.8 \41.4 & 43\end{bmatrix} \begin{bmatrix}100\110\end{bmatrix} = \begin{bmatrix}58.9 \times 100 + 62.8 \times 110 \41.4 \times 100 + 43 \times 110\end{bmatrix}\]

Step 4: Calculate Matrix Multiplication

Calculate the multiplication:1. For the first row, first element: \(58.9 \times 100 + 62.8 \times 110 = 5890 + 6908 = 12798\)2. For the second row, first element: \(41.4 \times 100 + 43 \times 110 = 4140 + 4730 = 8870\)So, \(A \times B = \begin{bmatrix}12798\8870\end{bmatrix}\).

Step 5: Interpret the Result

The resulting matrix \(AB\) \(\begin{bmatrix}12798 \cr 8870\end{bmatrix}\) provides the number of registered men and women voters as well as those who voted in millions.- The first element (12798) is the estimated number of registered voters, including men and women combined.- The second element (8870) is the estimated number of voters, again combined for both genders.

Key Concepts

Understanding U.S. Census Bureau Statistics

Statistics provided by the U.S. Census Bureau offer valuable insights into demographic data. In this exercise, the statistics provide specific details on the voting behavior of men and women over the age of 18 during the 2000 national election.

It is important to understand how to interpret these percentages:

• The percentage of registered voters reflects how many adults have signed up to vote, not how many actually vote. For instance, 58.9% of men were registered to vote.
• The percentage of actual voters among the registered tells us who participated in the voting process. For men, this was only 41.4%.

These statistics can help policymakers, researchers, and educators understand voter engagement or lack thereof during election periods. By investigating the trends over time, one can gain insights into growing or declining political engagement among different gender populations.

Analyzing Gender Representation in Voting

Gender representation in voting highlights how both men and women contribute to the electoral process. By using the statistics from the U.S. Census Bureau, we see clear distinctions in voting patterns among the genders:

• 62.8% of women were registered to vote, indicating a slightly higher registration rate than men.
• Despite this higher registration, a relatively small portion of women, specifically 43%, actually voted.

Understanding these figures is crucial for analyzing gender representation, as they reflect not just political participation but also reveal potential barriers that each gender might face when it comes to casting their votes. This way of analyzing data helps in identifying opportunities to increase participation and address specific issues that may prevent voters from engaging fully in the democratic process.

Precalculus Application of Matrices

The use of matrices in precalculus is a powerful method to manage and compute multi-variable data sets. In this exercise, we take information from voter statistics and apply matrix multiplication—a fundamental operation in linear algebra.

Here's how matrix multiplication works in this context:

• Matrix A contains percentages of registered voters and those who voted, organized by gender.
• Matrix B represents the total population of males (100M) and females (110M).
• By multiplying Matrix A and B, we accumulate meaningful data that combines these statistics with actual population figures.

Calculate the result as follows: \[ A \times B = \begin{bmatrix} 58.9 & 62.8 \ 41.4 & 43 \end{bmatrix} \begin{bmatrix} 100 \ 110 \end{bmatrix} = \begin{bmatrix} 12798 \ 8870 \end{bmatrix} \] This result interprets how many millions of both men and women were registered and actually voted, giving us real-world numbers that relate to the original percentages. Matrix applications like these are fundamental in precalculus, offering significant advantages in data interpretation.