Problem 43 - Applied Calculus

Question

The population of a city is 50,000 in 2008 and is growing at a continuous yearly rate of \(4.5 \%\) (a) Give the population of the city as a function of the number of years since \(2008 .\) Sketch a graph of the population against time. (b) What will be the city's population in the year \(2018 ?\) (c) Calculate the time for the population of the city to reach \(100,000 .\) This is called the doubling time of the population.

Step-by-Step Solution

Answer

(a) \( P(t) = 50000 e^{0.045t} \), (b) ~78,141, (c) ~15.4 years.

Step 1: Identifying the Population Growth Model

Since the city is experiencing continuous growth, we will use the exponential growth model. The formula for continuous growth is given by \( P(t) = P_0 e^{rt} \), where \( P_0 \) is the initial population, \( r \) is the growth rate, and \( t \) is the time in years.

Step 2: Writing the Population Function

Here, \( P_0 = 50000 \) and \( r = 0.045 \) (as 4.5% in decimal form). Therefore, the population function is \( P(t) = 50000 e^{0.045t} \), where \( t \) is the number of years since 2008.

Step 3: Calculating Population in 2018

Substitute \( t = 2018 - 2008 = 10 \) years into the population function: \( P(10) = 50000 e^{0.045 \times 10} \). Calculate to find the population at 2018.

Step 4: Graphing Population Over Time

To graph the population function, plot \( P(t) = 50000 e^{0.045t} \) on a graph, with \( t \) (years since 2008) on the horizontal axis and \( P(t) \) (population) on the vertical axis. The graph will be an exponential curve starting at 50,000 and rising as \( t \) increases.

Step 5: Calculating Doubling Time

We need to determine when the population reaches 100,000. Set \( P(t) = 100000 = 50000 e^{0.045t} \) and solve for \( t \). Dividing both sides by 50000 gives \( 2 = e^{0.045t} \). Take the natural logarithm of both sides to get \( \ln(2) = 0.045t \). Solve for \( t \) to find the doubling time.

Step 6: Conclusion

The population function is \( P(t) = 50000 e^{0.045t} \). In 2018, the population is approximately 78,141. The doubling time for the population is about 15.4 years.

Key Concepts

Continuous Compounding

Continuous compounding refers to the process of calculating interest or growth continuously over time, without breaks. In the context of population growth, this kicks in because populations can in theory increase at any instant. To express this mathematically, we use a formula that includes the number 'e,' which is approximately equal to 2.71828.
This base of the natural logarithm, 'e,' is important because it makes calculations involving exponential growth smooth and continuous.

To model the continuous growth of a population, the formula we use is:

• \( P(t) = P_0 e^{rt} \)

where:

• \( P(t) \) is the future population size.
• \( P_0 \) is the initial population size at the start.
• \( r \) is the continuous growth rate (in decimal form).
• \( t \) is the time in years since the start.

With continuous compounding, population growth doesn't happen at fixed intervals but continuously over time, making it a very useful model for predicting growth over a long period.

Population Doubling Time

Population doubling time is a measure of how long it takes for a population to double in size at a continuous growth rate. This helps understand the speed of growth and anticipate future population numbers without predicting exact numbers at every point in time.

To find the doubling time, use the formula related to continuous compounding:

• Set \( P(t) = 2P_0 \) in the exponential growth formula.
• \( 2P_0 = P_0 e^{rt} \)
• Canceling \( P_0 \) from both sides gives: \( 2 = e^{rt} \)
• Take the natural logarithm on both sides: \( \ln(2) = rt \)
• Solve for \( t \): \( t = \frac{\ln(2)}{r} \)

In practice, if a city has a continuous growth rate of 4.5%, the doubling time would be approximately 15.4 years. Understanding doubling time enables better planning and resource allocation to accommodate growth over time.

Exponential Growth Model

The exponential growth model is a way to describe how quantities increase rapidly over time. Unlike linear growth, where quantities grow by the same amount each time period, exponential growth refers to growth at a constant percent rate each time period.

This model is crucial for many fields:

• In biology, it explains the growth of populations.
• In finance, it describes compound interest.
• In physics, it can explain radioactivity decay patterns inversely.

In our specific case, using the exponential growth model helps us predict future population sizes given an initial population and a continuous growth rate. The general formula is:

• \( P(t) = P_0 e^{rt} \)

This model predicts not just population in 10 years, but also future trends over decades. When graphing this model, you'll see an upward-curving line, illustrating how quickly numbers grow as time passes. Understanding the exponential growth model helps us anticipate significant changes over time, especially in contexts where resources and planning depend on population sizes.