Problem 36 - 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

(i) Formula: \(P(t) = 1000(1.05)^t\); 10-year estimate: 1629. (ii) Formula: \(P(t) = 1000e^{0.05t}\); 10-year estimate: 1649.

Step 1: Understanding Annual Growth Rate

For an annual growth rate, the population at time \(t\) can be modeled using the formula for exponential growth: \( P(t) = P_0 (1 + r)^t \). Here, \(P_0\) is the initial population, \(r\) is the annual growth rate, and \(t\) is the time in years. We'll use this to find the formula for part (a)(i).

Step 2: Calculating Population with Annual Rate Formula

Using the formula \(P(t) = P_0 (1 + r)^t \) for an annual growth rate:- \(P_0 = 1000\)- \(r = 0.05\)So, the formula becomes \(P(t) = 1000 (1 + 0.05)^t = 1000 (1.05)^t\).

Step 3: Understanding Continuous Growth Rate

For a continuous growth rate, the population can be modeled with the formula:\( P(t) = P_0 e^{rt} \). In this case, \(e\) is the base of the natural logarithm, and \(r\) represents the continuous growth rate. We'll use this to find the formula for part (a)(ii).

Step 4: Calculating Population with Continuous Rate Formula

Using the formula \(P(t) = P_0 e^{rt}\) for a continuous growth rate:- \(P_0 = 1000\)- \(r = 0.05\)So, the formula becomes \(P(t) = 1000 e^{0.05t}\).

Step 5: Estimating Population in 10 years (Annual Rate)

For part (b),(i), use the formula from Step 2 with \(t = 10\):\(P(10) = 1000 (1.05)^{10}\).Calculating this, \(P(10) \approx 1000 \times 1.6289 = 1628.9\). Therefore, the estimated population is approximately 1629.

Step 6: Estimating Population in 10 years (Continuous Rate)

For part (b),(ii), use the formula from Step 4 with \(t = 10\):\(P(10) = 1000 e^{0.05 \times 10}\).This simplifies to \(P(10) = 1000 e^{0.5}\), and using the value of \(e^{0.5} \approx 1.6487\), we find \(P(10) \approx 1000 \times 1.6487 = 1648.7\). Therefore, the estimated population is approximately 1649.

Key Concepts

Annual Growth Rate

In mathematics, the annual growth rate is a measure used to model how a quantity, such as a population, grows over time. This model is particularly suitable for understanding growth that occurs on a yearly basis. The formula to compute annual growth is given by \( P(t) = P_0 (1 + r)^t \). Here:

• \(P(t)\) represents the population at time \(t\).
• \(P_0\) is the initial population size.
• \(r\) is the annual growth rate, expressed as a decimal. For example, \(5\%\) would be written as \(0.05\).
• \(t\) is the number of years into the future you want to predict.

The annual growth assumption assumes that growth compounds at the end of each year. Therefore, if a city's population starts at 1000 and grows at \(5\%\) annually, in 10 years, its population will be approximately 1629. Use the calculations \(P(10) = 1000(1.05)^{10}\) to achieve this estimate. It effectively demonstrates how exponential growth magnifies over time.

Continuous Growth Rate

The continuous growth rate considers growth as occurring constantly, rather than at discrete intervals. With this model, changes happen continuously, approximated by using the natural exponential function. The continuous growth formula is \( P(t) = P_0 e^{rt} \), where:

• \(P(t)\) stands for the population at time \(t\).
• \(P_0\) is the starting population size.
• \(r\) is the continuous growth rate.
• \(e\) is the base of the natural logarithm, approximately equal to 2.71828.
• \(t\) refers to time in years.

When a city has a population of 1000 with a continuous growth rate of \(5\%\), we calculate the population in 10 years using \( P(10) = 1000e^{0.05 \times 10} \). This leads to approximately 1649 people. This model is often used in natural processes as it smoothly captures ongoing changes without the step-like increments seen in discrete models.

Population Model

Population models are mathematical representations used to predict changes in population size over time. They help in understanding how various factors can affect population dynamics.In the context of exponential growth, population models focus on how the size of a population grows in relation to itself, making them a key tool in demographics.

• These models assume that the growth rate is proportional to the current size, meaning larger populations grow faster.
• Models can be discrete, like the annual growth rate, or continuous, depending on the situation.
• They are crucial in planning and decision-making for city planning, resource allocation, and understanding ecological systems.

For instance, when predicting a city's growth, understanding whether that growth is better represented by an annual or continuous model can significantly impact projections and plans. By applying relevant formulas, such as \( P(t) = P_0 (1 + r)^t \) or \( P(t) = P_0 e^{rt} \), planners can effectively estimate future resource needs and growth management strategies.