Problem 50 - Calculus: Early Transcendentals

Question

If the birth rate of a population is \( b(t) = 2200 e^{0.024t} \) people per year and the death rate is \( d(t) = 1460 e^{0.018t} \) people per year, find the area between these curves for \( 0 \le t \le 10 \). What does this area represent?

Step-by-Step Solution

Answer

The area represents the net growth of the population from year 0 to year 10.

Step 1: Understand the Problem

We are given functions for the birth rate \( b(t) = 2200 e^{0.024t} \) and the death rate \( d(t) = 1460 e^{0.018t} \). We need to find the area between these two curves from \( t = 0 \) to \( t = 10 \), which represents the net growth of the population.

Step 2: Set Up the Expression for the Area

The area between the two curves is given by the integral of the difference between the birth and death rates. We set up the integral as:\[\int_{0}^{10} (b(t) - d(t)) \, dt = \int_{0}^{10} (2200 e^{0.024t} - 1460 e^{0.018t}) \, dt.\]

Step 3: Solve the Integral

Integrate each term separately:1. \( \int 2200 e^{0.024t} \, dt = \frac{2200}{0.024} e^{0.024t} \)2. \( \int 1460 e^{0.018t} \, dt = \frac{1460}{0.018} e^{0.018t} \).Evaluate both integrals from \( t=0 \) to \( t=10 \).

Step 4: Calculate Definite Integrals

Evaluate the definite integrals:\[\left[ \frac{2200}{0.024} e^{0.024t} \right]_{0}^{10} = \frac{2200}{0.024} (e^{0.24} - 1),\]\[\left[ \frac{1460}{0.018} e^{0.018t} \right]_{0}^{10} = \frac{1460}{0.018} (e^{0.18} - 1).\]Calculate these values to find the solutions for each integral.

Step 5: Compute the Area and Interpret

Subtract the results from Step 4 to find the area:\[\frac{2200}{0.024} (e^{0.24} - 1) - \frac{1460}{0.018} (e^{0.18} - 1).\]This area represents the net population growth between \( t = 0 \) and \( t = 10 \) years.

Key Concepts

Integral Calculus

Integral Calculus is a fundamental branch of calculus focusing on the concept of integration. Imagine it as a way to sum up parts to find a whole. It's often used in various fields like physics, engineering, and economics to calculate areas, volumes, and more. In our exercise, we use integral calculus to find the area between two curves.
This area helps us understand complex phenomena, like population growth, by calculating the net change over time.

• Integration: The process of finding integrals involves determining the anti-derivative or the reverse process of differentiation.
• Applications: Integral calculus helps solve problems involving accumulation of quantities and determining areas between curves.
• Techniques: There are various techniques for integration, including substitution and integration by parts.

Understanding these concepts can help decode real-world problems like the one we're tackling here.

Population Growth

Population growth examines how populations increase over time. In terms of mathematics, it's a process often modeled by functions like those in our exercise. Insightful as it is, the function given gives a clear picture of how a population might naturally progress or decline.
In mathematical terms:

• Birth Rate: The rate at which individuals are added to a population. In our problem, represented by the function \( b(t) = 2200 e^{0.024t} \).
• Death Rate: The rate at which individuals leave a population. Here, it's \( d(t) = 1460 e^{0.018t} \).
• Net Growth: Calculated by subtracting the death rate from the birth rate, providing a clear picture of overall growth.

By analyzing these functions, one can make solid predictions about population trends.

Exponential Functions

Exponential functions are crucial in modeling growth and decay in numerous situations, especially in our exercise about population growth. They work by raising a constant (like Euler's number \( e \)) to a power, which is usually a variable. This type of function describes rapid increases or decreases over time.
Key aspects include:

• Form: Typically written as \( f(t) = a \, e^{kt} \), where \( a \) is the initial amount, \( e \) is approximately 2.718, and \( k \) is the growth or decay rate.
• Applications: Used extensively in population growth, finance, radioactive decay, and more.
• Characteristic: The function shows continuous growth or decline, making it suitable for natural phenomena.

Understanding exponential functions can shine a light on how quickly variables can change under constant growth rates.

Definite Integral

A definite integral is a tool used in calculus to calculate the net area under a curve over a specific interval. It's vital for finding accumulated quantities like distance, area, or, in this case, population growth over a time period.
Consider these points:

• Concept: Involves taking the integral of a function between two limits, \( a \) and \( b \). Mathematically, expressed as \( \int_{a}^{b} f(t) \, dt \).
• Graphically: Represents the total area between the graph of the function and the x-axis over the interval [\( a, b \)].
• Application in Exercise: We compute the integral of the difference between birth and death rates over the interval \( t = 0 \) to \( t = 10 \), providing the overall growth in population.

By mastering definite integrals, we gain powerful insight into cumulative totals that demonstrate change over time.