Problem 17 - Precalculus

Question

In 2003 an estimated 1 million people had been infected with HIV in the United States. If the infection rate increases at an annual rate of \(2.5 \%\) a year compounding continuously, how many Americans will be infected with the HIV virus by \(2010 ?\)

Step-by-Step Solution

Answer

Approximately 1,191,000 Americans will be infected by 2010.

Step 1: Understand the Problem

We need to find the number of people infected with HIV by 2010, given that in 2003 there were 1 million infected and the infection rate increases continuously at 2.5% per year.

Step 2: Identify the Formula for Continuous Growth

When dealing with continuous growth, use the formula: \( P(t) = P_0 \times e^{rt} \) where \( P_0 \) is the initial amount, \( r \) is the rate of growth, \( t \) is the time in years, and \( e \) is the base of the natural logarithm.

Step 3: Define the Variables

Here, \( P_0 = 1,000,000 \) (initial population in 2003), \( r = 0.025 \) (2.5% growth rate), and \( t = 2010 - 2003 = 7 \) years.

Step 4: Plug Values into the Formula

Substitute the values into the continuous growth formula: \( P(7) = 1,000,000 \times e^{0.025 \times 7} \).

Step 5: Calculate the Result

First, calculate the exponent: \( 0.025 \times 7 = 0.175 \). Then, use a calculator to find \( e^{0.175} \), which is approximately 1.191. Multiply this by 1,000,000 to get \( 1,191,000 \) people.

Key Concepts

Continuous Compounding

When we talk about continuous compounding in mathematics, we refer to a scenario where growth occurs constantly at every possible instant. This is unlike ordinary compounding, such as annually or monthly, where growth adds up at regular intervals. An easy way to picture this is to think of continuous compounding as the most frequent possible form of compounding.
It uses the formula:

• \( P(t) = P_0 \times e^{rt} \)
• \(P_0\) is the initial amount
• \(r\) is the rate of growth
• \(t\) is time in years
• \(e\) is the base of natural logarithms

This formula helps us calculate the future value of an investment, or in this case, the future infected population. We use the power of the mathematical constant \(e\) to grow the initial amount \(P_0\) continuously, ensuring accuracy in predictions for things like interest or, as in our example, HIV infection rates.

HIV Infection Modeling

HIV infection modeling is a mathematical approach used to predict the spread of HIV within a population. Such models are vital for understanding how the virus spreads over time and are particularly useful in planning healthcare resources and strategies for prevention.
In the case of the United States HIV infection rates, the model can predict future numbers of infected individuals by using known data, like current rates and initial populations, while applying mathematical formulas for growth. With a given initial population and a continuous compounding growth rate, we can assess expected future infections over several years.
Such models use the exponential growth formula to calculate the increase based on a given percentage rate over a specified period. This allows public health officials to better plan interventions, measure the effectiveness of current strategies, and estimate future healthcare requirements.

Natural Logarithm Base

The natural logarithm base, denoted by \(e\), is approximately equal to 2.718. This mathematical constant is essential in calculations involving exponential growth and decay. Our formula for continuous compounding relies heavily on \(e\), allowing for the representation of situations where quantities grow continuously, rather than in discrete steps.
The use of the natural logarithm base comes from solving calculus problems involving growth rates, providing a seamless way to model real-life processes like population growth, radioactive decay, and viral infections. A key feature of \(e\) is its natural properties, where the derivative of \(e^x\) is equal to \(e^x\), making it unique among functions.
In the context of our problem, using the base \(e\) allows us to accurately calculate future values based on a continuous growth process, essential for precise predictions needed in understanding HIV spread.