Problem 50 - A Graphical Approach to College Algebra

Question

In \(2012,63 \%\) of the U.S. population was non-Hispanic white, and this number is expected to be \(43 \%\) in \(2060 .\) (Source: U.S. Census Bureau.) (a) Find \(C\) and \(a\) so that \(P(x)=C a^{x-2012}\) models these data, where \(P\) is the percent of the population that is non-Hispanic white and \(x\) is the year. Why is \(a<1 ?\) (b) Estimate \(P\) in 2020 (c) Use \(P\) to estimate when \(50 \%\) of the population could be non-Hispanic white.

Step-by-Step Solution

Answer

(a) \( C = 63, a \approx 0.9917 \). (b) \( P(2020) \approx 59.97\% \). (c) \( x \approx 2044 \).

Step 1: Understanding the Model Equation

The model equation given is \( P(x) = C a^{x-2012} \), where \( P(x) \) represents the percentage of the population that is non-Hispanic white in year \( x \). Here, \( C \) is the initial percentage in 2012, and \( a \) is the decay factor.

Step 2: Setting Up Equation for 2012

In 2012, the equation is \( P(2012) = C = 63 \). This directly comes from the problem statement that 63% of the population was non-Hispanic white in 2012.

Step 3: Setting Up Equation for 2060

For the year 2060, the equation is \( P(2060) = C a^{2060-2012} = 43 \). Substituting the value of \( C \) from the previous step, we get: \( 63 a^{48} = 43 \).

Step 4: Solving for \( a \)

Solve the equation \( 63 a^{48} = 43 \) for \( a \), which gives \( a^{48} = \frac{43}{63} \). Hence, \( a = \left(\frac{43}{63}\right)^{\frac{1}{48}} \). Calculating, we find \( a \approx 0.9917 \).

Step 5: Explaining Why \( a < 1 \)

The value of \( a < 1 \) indicates a decay or decrease over time, which matches the context of the problem where the percentage is expected to decrease.

Step 6: Estimating \( P \) in 2020

For 2020, use the model equation: \( P(2020) = 63 \times \left(0.9917\right)^{8} \). Calculating this gives \( P(2020) \approx 59.97 \% \).

Step 7: Estimating When \( P(x) = 50 \)

To find when \( P(x) = 50 \), use the equation: \( 63 (0.9917)^{x-2012} = 50 \). Solving for \( x \), first solve \( (0.9917)^{x-2012} = \frac{50}{63} \). Taking the logarithm of both sides gives: \( (x-2012) \log(0.9917) = \log\left(\frac{50}{63}\right) \). Solving gives \( x \approx 2044 \).

Key Concepts

Percent of Population

The percent of the population is a crucial element in understanding demographic shifts over time. In demographic studies, the percent of the population often refers to a specific subset of the population, like in this case, the non-Hispanic white population in the United States over several decades.
In 2012, it was recorded that 63% of the U.S. population was non-Hispanic white. This percentage serves as the initial point for the exponential decay model. Later on, by 2060, this is projected to decrease to 43%.
This means that over time, the proportion of non-Hispanic whites in the overall population is declining. This is a perfect scenario to employ the exponential decay model because it helps forecast how a percentage will change year by year based on historical data. This model is especially helpful in predicting long-term trends and planning based on the expected demographic changes.

Decay Factor

The decay factor is a key component in the exponential decay model and reflects a decrease in a certain quantity over time. In this context, the decay factor, represented by the letter 'a' in the exponential model, indicates a gradual decrease in the non-Hispanic white population over the years.
Mathematically, the decay factor is determined through the equation derived from the model: \( P(x) = C \times a^{x-2012} \). A decay factor less than 1, as is applicable here, suggests that the percentage decreases with time.

• The initial equation \( 63a^{48} = 43 \) was solved via the calculation \( a^{48} = \frac{43}{63} \).
• Solving for 'a' gives us \( a \approx 0.9917 \).

The fact that 'a' is less than 1 confirms that the model reflects a decay scenario, where the group's percentage in the overall population declines each year.

U.S. Census Data

U.S. Census data provides reliable insights into the demographic changes and trends throughout the country. The Census Bureau gathers critical information about the population, including ethnic breakdowns, which allows for informed analysis and forecasting.
For instance, the non-Hispanic white population percentage data from the U.S. Census is used to model future demographic changes using an exponential decay formula. This data forms the backbone of demographic models providing historical and projected information.
This data not only supports demographic studies but can also be fundamental for

• policy-making,
• urban planning,
• resource allocation,
• social services design.

Understanding how certain population groups will change over time can help various sectors prepare for future societal needs.