Problem 22 - Applied Calculus

Question

A city's population is 1000 and growing at \(5 \%\) a year. (a) Find a formula for the population at time \(t\) years from now assuming that the \(5 \%\) per year is an: (i) Annual rate (ii) Continuous annual rate (b) In each case in part (a), estimate the population of the city in 10 years.

Step-by-Step Solution

Answer

Annual growth estimates the population at 1629. Continuous growth estimates it at 1649.

Step 1: Understand the Annual Growth Formula

When the growth is expressed as an annual rate, the population grows geometrically each year. The formula to calculate the population after \(t\) years with an annual growth rate of "\(r\)" is \(P(t) = P_0(1 + r)^t\). In this context, \(P_0 = 1000\) and \(r = 0.05\) (since \(5\% = 0.05\)).

Step 2: Apply the Formula for Annual Growth

Substitute the values into the formula. \[P(t) = 1000(1 + 0.05)^t\]This formula represents the population at any year \(t\) assuming \(5\%\) is an annual rate.

Step 3: Understand the Continuous Growth Formula

When the growth is expressed as a continuous rate, the population grows exponentially over time. The formula used is \(P(t) = P_0 e^{rt}\), where \(e\) is the mathematical constant approximately equal to 2.71828, and \(r\) is the continuous growth rate. Here, \(P_0 = 1000\) and \(r = 0.05\).

Step 4: Apply the Formula for Continuous Growth

Substitute the values into the formula for continuous growth:\[P(t) = 1000 e^{0.05t}\]This formula gives the population at any year \(t\) assuming \(5\%\) is a continuous annual growth rate.

Step 5: Calculate Population After 10 Years with Annual Growth

Substitute \(t = 10\) into the annual growth formula:\[P(10) = 1000(1 + 0.05)^{10} = 1000(1.05)^{10}\]Calculate \((1.05)^{10}\) to find \(P(10) \approx 1000 \times 1.6289 = 1628.89\).

Step 6: Calculate Population After 10 Years with Continuous Growth

Substitute \(t = 10\) into the continuous growth formula:\[P(10) = 1000 e^{0.5} = 1000 \times 2.71828^{0.5}\]Calculate \(e^{0.5}\) which is approximately 1.6487. Therefore, \(P(10) \approx 1000 \times 1.6487 = 1648.72\).

Key Concepts

Annual Growth Rate

The annual growth rate is a way of describing how a quantity increases over time when it grows by a fixed percentage each year. In the context of population growth, it means that the number of inhabitants increases by a certain percentage each year.
For example, a city with a population of 1000 people growing by 5% annually will use the annual growth formula to predict future population sizes. The formula used here is:- \( P(t) = P_0(1 + r)^t \) - \( P_0 \) is the initial population, which is 1000 in this scenario. - \( r \) is the annual growth rate, expressed as a decimal (5% becomes 0.05). - \( t \) is the time in years.
Using this formula for 10 years:- \( P(10) = 1000(1 + 0.05)^{10} \)- Calculating further, you find that \((1.05)^{10} \approx 1.6289\), thus producing a population of approximately 1628.89 after 10 years.
Understanding this concept helps predict how populations might change over time with consistent growth.

Continuous Growth Rate

Continuous growth rate is slightly different from the annual growth rate. It assumes growth is happening constantly, at every possible moment, rather than just at the end of each year. This tends to provide a slightly higher population size due to the continuous compounding of growth.
The formula for continuous growth is:- \( P(t) = P_0 e^{rt} \) - \( e \) is the base of natural logarithms, approximately equal to 2.71828. - \( r \) remains the growth rate as a decimal, in this case, 0.05. - \( t \) is still the time in years.
When we apply this formula over 10 years, we calculate:- \( P(10) = 1000 e^{0.05 imes 10} \)- Solving this, \( e^{0.5} \approx 1.6487 \), results in a future population of approximately 1648.72.
The process of continuous growth allows for a more accurate reflection of complex real-world trends, often seen in natural processes and finance.

Population Growth Formula

Population growth formulas offer the tools we need to calculate future population sizes based on different growth models. Both the annual and continuous growth models provide insights into how populations change, but they are used in slightly different scenarios. Here's a summary of both:

• Annual Growth Formula: Used when growth is compounded at the end of each year. It's simple and suitable for many applications where monitoring annual change is sufficient.
• Continuous Growth Formula: Applied where growth compounds continually. It's especially useful for modeling scenarios where growth happens at very short intervals.

Different contexts may call for either the annual or continuous model, depending on how growth is realized in reality. Understanding these formulas helps predict and accommodate for future resource needs, urban planning, and numerous other applications. They are crucial in demographics, ecology, and economics, among other fields. In any situation where growth is a factor, choosing the right formula will make all the difference in forecasting and planning.