Problem 32 - Algebra A Combined Function

Question

Solve. The number of victims of a flu epidemic is increasing according to the formula \(y=y_{0} e^{0.075 t}\). In this formula, is time in weeks and \(y_{0}\) is the given population at time 0 . If 20,000 people are currently infected, how many might be infected in 3 weeks? Round to the nearest whole number.

Step-by-Step Solution

Answer

About 25,046 people might be infected in 3 weeks.

Step 1: Understand the Problem

We need to determine the number of people infected after 3 weeks, given the initial number of infected people is 20,000. The model of increase is given by the formula \(y = y_0 e^{0.075t}\).

Step 2: Identify Known Values

From the problem statement, we know that \(y_0 = 20000\) and \(t = 3\) weeks.

Step 3: Substitute Known Values into the Formula

Plug the known values into the formula: \(y = 20000 \cdot e^{0.075 \cdot 3}\).

Step 4: Calculate the Exponent Component

Calculate the exponent: \(0.075 \times 3 = 0.225\).

Step 5: Compute the Exponential Value

Evaluate the exponential: \(e^{0.225} \approx 1.25232\).

Step 6: Find the Estimated Number of Infected People

Calculate \(y = 20000 \times 1.25232\). This gives \(y \approx 25046.4\).

Step 7: Round to the Nearest Whole Number

Since we need a whole number, round \(25046.4\) to \(25046\).

Key Concepts

Mathematical Modeling

Mathematical modeling is a useful tool to represent real-world scenarios with mathematical formulas. It helps us understand and predict the behavior of systems over time. In this exercise, the focus is on modeling how a flu epidemic spreads among a population. The formula used is of exponential growth nature: \[y = y_0 e^{0.075t}\]where:

• \(y_0\) represents the initial number of people infected,
• \(t\) is the time in weeks,
• \(e\) is the base of natural logarithms, approximately equal to 2.71828,
• and 0.075 is the growth rate.

Exponential growth occurs when the rate of increase in a population is proportional to its current value. This formula is a classic example of mathematical modeling, capturing how quickly an illness can spread when unchecked.

Flu Epidemic

An epidemic occurs when there is a rapid spread of an infectious disease to a large number of individuals within a short period. In the context of a flu epidemic, the disease can spread quickly from person to person, reflecting exponential growth patterns seen in many natural processes. The flu is caused by a virus that is highly contagious. It can spread through:

• Coughing and sneezing which project droplets with the virus into the air.
• Touching surfaces contaminated with the virus and then touching one's face.
• Close contact with infected individuals.

Mathematical modeling of flu epidemics allows us to predict the number of infections over time, helping communities and health organizations plan responses and resources better. Vaccination and health measures can alter the course of an epidemic, changing the parameters of the model.

Infected Population Calculation

Calculating the infected population using the given formula involves following a straightforward process. Here, the formula \[y = y_0 e^{0.075t}\]lets us input known initial conditions to find how much the infected population will grow.To solve the exercise:1. **Identify Initial Values:** - Initial infected population, \(y_0 = 20000\). - Time period, \(t = 3\) weeks. 2. **Substitute Known Values:** - Insert into the formula: \(y = 20000 \times e^{0.075 \times 3}\). 3. **Calculate Exponential Growth:** - Compute \(e^{0.225}\) which gives approximately 1.25232. 4. **Multiply Values:** - Calculate \(y = 20000 \times 1.25232 = 25046.4\).5. **Round to Nearest Whole Number:** - Results in approximately 25046 infected individuals after 3 weeks.Using this process, we identify potential future impacts of the epidemic and strategize accordingly.