Question

The population of a certain country is known to increase at a rate proportional to the number of people presently living in the country. If after two years the population has doubled, and after three years the population is 20,000 , estimate the number of people initially living in the country.

Short answer

Answer: The estimated initial population of the country is around 10,000 people.

Step-by-step solution

Write the exponential growth equation

The exponential growth equation is usually given in the form of: P(t) = P₀ * e^(k * t) Where: - P(t) is the population at time t - P₀ is the initial population - e is the base of the natural logarithm (approximately 2.718) - k is the exponential growth constant - t is the number of years

Double the population after two years

The problem states that the population doubles after two years. We can use this information and the exponential growth equation to express the population after two years. P(2) = 2 * P₀ Since P(t) = P₀ * e^(k * t), then P(2) is: 2 * P₀ = P₀ * e^(2k)

Calculate k

From the equation in Step 2, we can calculate the value of k. 2 * P₀ = P₀ * e^(2k) Divide both sides by P₀ to isolate e^(2k): 2 = e^(2k) Now, take the natural logarithm of both sides to solve for k: ln(2) = 2k Divide by 2 to get the value of k: k = ln(2) / 2

Population after three years

Now that we have the value of k, we can use the exponential growth equation to calculate the initial population, P₀. We know that the population after three years, P(3), is 20,000. 20,000 = P₀ * e^(3k) Substitute the value of k from Step 3: 20,000 = P₀ * e^(3 * (ln(2) / 2))

Estimate the initial population P₀

Now, solve for the initial population, P₀: P₀ = 20,000 / e^(3 * (ln(2) / 2)) P₀ ≈ 10,000 The estimated initial population of the country is around 10,000 people.

Key concepts

Integral Calculus

Integral calculus, a fundamental component of calculus, involves finding the area under a curve on a graph, often referred to as integrating a function. It's the reverse process of differentiation, where we find the rate at which a function is changing at any point. In the context of exponential growth, integral calculus is used to determine total growth over a period. However, for the exponential growth equation provided in the problem, we are more focused on finding the exponential growth constant and the initial quantity, rather than accumulating quantities over time, which is the primary domain of integral calculus. Nonetheless, understanding integration is essential for a comprehensive grasp of calculus and its applications in various fields such as physics, engineering, and economics.

Natural Logarithm

The natural logarithm, denoted as 'ln', is the logarithm to the base 'e', where 'e' is an irrational and transcendental number approximately equal to 2.718. The natural logarithm of a number 'x' answers the question: 'To what power must the number 'e' be raised, to obtain the value 'x'?' It's a critical concept in solving equations involving 'e', like the exponential growth equation. In the textbook problem, we saw how taking the natural logarithm of both sides of an equation with an exponent that includes 'e' simplifies the equation to allow for the isolation and solving for the growth constant 'k'. Understanding the properties of natural logarithms is essential for solving many types of exponential problems across various scientific disciplines.

Exponential Growth Constant

The exponential growth constant 'k' is a crucial part of the exponential growth equation, symbolizing the rate of growth. When 'k' is positive, the function represents exponential growth, and conversely, a negative 'k' indicates exponential decay. The value of 'k' determines how quickly the quantity grows or decays over time. In our textbook example, the value of 'k' was found using the information that the population doubles in two years, leading to the equation '2 = e^(2k)'. After applying natural logarithms, we find 'k' to be 'ln(2)/2'. This constant is essential for predicting future population sizes and is used extensively in population models, financial projections, and natural phenomena such as radioactive decay. Recognizing and calculating the growth constant is fundamental to effectively apply exponential growth models in real-world scenarios.