Problem 84 - Contemporary Precalculus

Question

The population of St. Petersburg, Florida (in thousands) can be approximated by the function $$ g(x)=-127.9+81.91 \ln x \quad(x \geq 70) $$ where \(x=70\) corresponds to 1970 (a) Estimate the population in 1995 and 2003 . (b) If this model remains accurate, when will the population be \(260,000 ?\)

Step-by-Step Solution

Answer

Question: Based on the given function \(g(x) = -127.9 + 81.91 \ln x\), estimate the population of St. Petersburg, Florida in 1995 and 2003, and find the year when the population will be 260,000. Answer: The estimated population in 1995 is about 246,113 people, and in 2003 is about 252,740 people. The population will reach 260,000 around the year 2009.

Step 1: STEP 1: Analyzing the function

Since \(x=70\) corresponds to 1970, we can find the values of \(x\) for 1995 and 2003 by calculating how many years have passed since 1970. 1995 - 1970 = 25 2003 - 1970 = 33 Therefore, \(x = 95\) for 1995, and \(x = 103\) for 2003.

Step 2: STEP 2: Estimating the population in 1995

Next, we'll plug in \(x = 95\) into the given function to estimate the population in 1995: \(g(95) = -127.9 + 81.91 \ln(95)\) Now, we can estimate the population in 1995: \(g(95) ≈ -127.9 + 81.91 \cdot 4.553877 ≈ 246.113\) (in thousands) So the estimated population in 1995 is about 246,113 people.

Step 3: STEP 3: Estimating the population in 2003

We'll plug in \(x = 103\) into the given function to estimate the population in 2003: \(g(103) = -127.9 + 81.91 \ln(103)\) Now, we can estimate the population in 2003: \(g(103) ≈ -127.9 + 81.91 \cdot 4.634729 ≈ 252.740\) (in thousands) So the estimated population in 2003 is about 252,740 people. #b) Year when the population will be 260,000#

Step 4: STEP 1: Setting up the equation for x

We want to find the value of \(x\) when the population is 260,000. Since the function represents the population in thousands, we need to rewrite 260,000 as 260 (thousands): \(-127.9 + 81.91 \ln x = 260\)

Step 5: STEP 2: Solving for x

Now, we'll solve for \(x\): \(81.91 \ln x = 387.9\) \(\ln x = \frac{387.9}{81.91}\) \(x = e^{\frac{387.9}{81.91}} ≈ 108.826\)

Step 6: STEP 3: Converting x to a year

Recall that \(x\) represents the number of years since 1900, and we found that \(x ≈ 108.826\). To find the year when the population will be 260,000, we simply add 108.826 years to 1900: \(1900 + 108.826 ≈ 2008.826\) This model predicts that the population of St. Petersburg, Florida will reach 260,000 around the year 2009 (since we can only consider whole years).

Key Concepts

Logarithmic Functions

Logarithmic functions are mathematical operations that help reverse exponentiation. If you know that a number results from raising another number (the base) to a certain power, a logarithm can help you determine the power. For instance, in the population model provided for St. Petersburg, the population is expressed using a logarithmic function:

• The function is denoted as \( g(x) = -127.9 + 81.91 \ln(x) \). It shows the relationship between a number of years since 1900 and the population size.
• The natural logarithm (\( \ln \)) is used, which is based on the constant \( e \approx 2.71828 \).
• This natural logarithm describes how the rate of change (in this case, population growth) decays or grows over time.

This logarithmic model helps estimate population at different times, using the collected data from past knowledge to predict future growth. It's effective because it scales large numbers down to manageable values for calculations, simplifying complex exponential phenomena.

Exponential Growth

Exponential growth describes processes where growth compounds over time; each step in time results in the more substantial accumulation or increase. In real-life situations like population growth, this is because each new generation builds on the previous ones.

• In our example, the population of St. Petersburg is modeled without directly using an exponential formula, but logarithmic growth often mirrors its effects indirectly through the constant rate of growth.
• This population growth can be observed from the years 1970 to 2003, where a compounded increase is estimated.

The idea is that with each passing year, the increments consider the current population, potentially resulting in a rapid swelling of numbers over a long timeline. The logarithmic expression in the model inherently accounts for this phenomenon by using the equation constants to control growth intensity.

Population Estimation

Population estimation using mathematical models helps project numbers when census data isn't available or to plan for future logistical needs. This estimation integrates mathematical understanding with historical data to create a plausible projection of future populations.

• The steps seen in the solution involve inputting the desired year into the logarithmic function and solving for \( g(x) \) to get an approximated number of thousands in the population.
• For example, solving \( g(95) \) results in an estimated population of about 246,113 people for the year 1995, demonstrating how the function interprets future population size based on historical trends.
• Similarly, the equation can be rearranged to solve for the year when the population will reach a certain threshold, like 260,000 people.

By modeling population with these techniques, planners and policy makers can make informed decisions regarding infrastructure, resources, and services to adequately accommodate future growth.