Question

A city is hit by an Asian flu epidemic. Officials estimate that \(t\) days after the beginning of the epidemic the number of persons sick with the flu is given by \(p(t)=120 t^{2}-2 t^{3}\), when \(0 \leq t \leq 40 .\) At what rate is the flu spreading at time \(t=10 ; t=20 ; t=40 ?\)

Short answer

Rates are 1800, 2400, and 0 people per day at times 10, 20, and 40, respectively.

Step-by-step solution

Understanding the Problem

We are given a function describing the number of people sick with the flu over time: \( p(t) = 120t^2 - 2t^3 \) for \( 0 \leq t \leq 40 \). The task is to find the rate at which the flu is spreading at specific times \( t = 10, 20, \text{and } 40 \). This requires calculating the derivative \( p'(t) \), which represents the rate of change.

Finding the Derivative

To find the rate at which the flu is spreading, we need the derivative of \( p(t) = 120t^2 - 2t^3 \). By applying the power rule, we have: \( p'(t) = \frac{d}{dt}(120t^2) - \frac{d}{dt}(2t^3) \). Calculate each derivative: \( \frac{d}{dt}(120t^2) = 240t \) and \( \frac{d}{dt}(2t^3) = 6t^2 \). Thus, \( p'(t) = 240t - 6t^2 \).

Calculating the Derivative at t = 10

Substitute \( t = 10 \) into the derivative to find the rate of spread: \( p'(10) = 240(10) - 6(10)^2 = 2400 - 600 = 1800 \). So, at \( t = 10 \), the flu is spreading at a rate of 1800 persons per day.

Calculating the Derivative at t = 20

Substitute \( t = 20 \) into the derivative to find the rate of spread: \( p'(20) = 240(20) - 6(20)^2 = 4800 - 2400 = 2400 \). Thus, at \( t = 20 \), the flu is spreading at a rate of 2400 persons per day.

Calculating the Derivative at t = 40

Substitute \( t = 40 \) into the derivative: \( p'(40) = 240(40) - 6(40)^2 = 9600 - 9600 = 0 \). At \( t = 40 \), the rate is 0, indicating the flu is no longer spreading.

Key concepts

Rate of Change

The rate of change is a key concept in calculus and mathematics in general. It helps us understand how a quantity alters over time. In the context of the exercise, the rate of change is used to determine how quickly the number of flu cases increases in the city over a specific time period. We use derivatives to calculate this rate.
To calculate the rate of change, one must derive the function in question. Here, the function is given as \( p(t) = 120t^2 - 2t^3 \). The derivative \( p'(t) \) provides the rate at which the flu cases are changing at any given time \( t \).

• The rate of change at \( t = 10 \) is 1800 persons per day.
• At \( t = 20 \), the rate increases to 2400 persons per day.
• By \( t = 40 \), the rate reaches 0, indicating no new cases.

Understanding the rate of change is crucial for making predictions and decisions based on data trends.

Power Rule

The power rule is a basic but powerful tool in calculus for finding derivatives of functions involving powers of a variable. This rule states that if you have a function of the form \( f(x) = ax^n \), the derivative \( f'(x) \) is calculated as \( anx^{n-1} \).
This rule simplifies the differentiation process, especially for polynomial functions, such as \( p(t) = 120t^2 - 2t^3 \). Applying the power rule here, we derive the function to get \( p'(t) = 240t - 6t^2 \), which represents the rate of change of the flu epidemic.
The power rule allows us to break down complex expressions easily, so we can understand and solve calculus problems like the one in this exercise more efficiently.

Epidemic Modeling

Epidemic modeling involves using mathematical models to understand and predict the behavior of infectious diseases within a population. This exercise focuses on modeling the spread of an epidemic, such as Asian flu, in a city.
By modeling the epidemic using the function \( p(t) = 120t^2 - 2t^3 \), we can understand how the number of infected individuals changes over time. Such models are crucial for public health officials to plan responses to outbreaks.
Understanding how quickly a disease spreads, when it will peak, and when it will start to decline can help in:

• Allocating resources effectively.
• Implementing social distancing measures.
• Planning vaccination campaigns.

Modeling epidemics with calculus allows for more informed decision-making during a health crisis.

Calculus Problem Solving

Calculus problem solving involves breaking down a problem into manageable steps, using mathematical techniques to find solutions. In this exercise, we solved the problem of determining how fast a flu epidemic is spreading using calculus methods.
Here are the steps involved in solving such a problem:

• Understanding the problem statement.
• Writing down the given function, \( p(t) = 120t^2 - 2t^3 \).
• Applying differentiation techniques, such as the power rule, to find the derivative \( p'(t) \).
• Substituting specific values of \( t \) to find the rate of change at different times.

By following these steps, students can learn to tackle similar calculus problems effectively, gaining a deeper understanding of the dynamics involved in changing quantities.