Problem 46 - Applied Calculus

Question

The population, \(P\), of a city (in thousands) at time \(t\) (in years) is \(P=700 e^{0.035 t} .\) Estimate the relative rate of change of the population at \(t=3\) using (a) \(\Delta t=1\) (b) \(\Delta t=0.1\) (c) \(\Delta t=0.01\)

Step-by-Step Solution

Answer

For (a) \( \Delta t=1 \), estimate: 0.0351; for (b) \( \Delta t=0.1 \), estimate: 0.0350; for (c) \( \Delta t=0.01 \), estimate: 0.0350006.

Step 1: Identify the Function and Variables

The population of the city is given by the function \( P(t) = 700 e^{0.035t} \), where \( P(t) \) is measured in thousands, and \( t \) is the time in years. We are to estimate the relative rate of change at \( t = 3 \) years.

Step 2: Formula for Relative Rate of Change

The relative rate of change at a time \( t \) is given by the formula \( \frac{P(t+\Delta t) - P(t)}{P(t)} \). Here we will estimate this relative change for different values of \( \Delta t \).

Step 3: Compute Relative Rate of Change for \( \Delta t = 1 \)

Substitute \( \Delta t = 1 \) into the formula. Calculate \( P(4) = 700 e^{0.035 \times 4} \) and \( P(3) = 700 e^{0.035 \times 3} \). Then find \( \frac{P(4) - P(3)}{P(3)} \).

Step 4: Compute Relative Rate of Change for \( \Delta t = 0.1 \)

Substitute \( \Delta t = 0.1 \) into the formula. Calculate \( P(3.1) = 700 e^{0.035 \times 3.1} \) and \( P(3) = 700 e^{0.035 \times 3} \). Then find \( \frac{P(3.1) - P(3)}{P(3)} \).

Step 5: Compute Relative Rate of Change for \( \Delta t = 0.01 \)

Substitute \( \Delta t = 0.01 \) into the formula. Calculate \( P(3.01) = 700 e^{0.035 \times 3.01} \) and \( P(3) = 700 e^{0.035 \times 3} \). Then find \( \frac{P(3.01) - P(3)}{P(3)} \).

Key Concepts

Relative Rate of Change

The concept of relative rate of change is crucial in understanding how a quantity changes over time relative to its current size. It is especially useful in analyzing growth processes, such as population growth.

When we talk about relative rate of change, we refer to the change in the population relative to its size at a certain point in time. The formula for relative rate of change is:

• \[\frac{P(t+\Delta t) - P(t)}{P(t)}\]

This represents how much the population changes within a small time period \( \Delta t \), divided by the population size at time \( t \).

By calculating this for different values of \( \Delta t \) (like 1, 0.1, or 0.01), we can see how the relative increase becomes more precise as the interval becomes smaller. This shows us how quickly the population is changing at that specific time.

Population Growth Model

A population growth model helps us understand and predict population size over time using mathematical equations.

In this exercise, the model is given by \( P(t) = 700 e^{0.035t} \). This is an exponential growth model, which is often used because populations tend to grow by a constant percentage.

• Exponential Growth: Populations grow exponentially when they increase at a consistent rate over time, which means they grow faster as they become larger.
• Understanding "e": The constant \( e \) (approximately 2.718) is the base of the natural logarithm, often used in calculations of growth because it provides a consistent rate of change over time.

In this model, 700 represents the initial population size in thousands, and 0.035 stands for the continuous growth rate per year. By entering different years into the model, we can see how the city's population might grow from this initial size.

Applied Calculus

Applied calculus is vital in solving real-world problems, like estimating how populations change over time.

In this situation, calculus helps us approximate how the population changes at a given point in time through calculations of the derivative and relative rates of change.

• Derivative: In calculus, the derivative represents the rate of change of a function. While not explicitly calculated in this exercise, it underlies the approximation of relative rates when \( \Delta t \) becomes very small.
• Linear Approximation: As \( \Delta t \) approaches zero, our calculation of the relative change approaches the derivative of \( P(t) \), providing a linear approximation of how fast the population grows at that moment.

By applying these principles of calculus, we can solve complex problems like population growth and make informed predictions.