91Ó°ÊÓ

When modeling a flu epidemic, one important aspect is to quantify the number of people affected over time. This type of modeling uses functions to describe how the disease spreads. In our exercise, we use a function, known as a differential equation, to model the rate at which new cases occur each day. Here, the rate of change of the sick population, represented as \( P'(t) \), is given by a quadratic equation: \( P'(t) = 120t - 3t^2 \). This means that as time \( t \) progresses, the number of new infections varies depending on the values of \( t \). For practical applications, it's crucial to find the total number of people sick at any given time \( t \), which means we need to find the function \( P(t) \). To achieve this, we need techniques from calculus, especially integration.

Integration is a fundamental concept in calculus used to find functions when given their derivatives. In this exercise, we're given the derivative \( P'(t) = 120t - 3t^2 \) and need to find the original function \( P(t) \). The process involves finding the indefinite integral of the given derivative.

To do this, integrate each term separately:
\[ P(t) = \int (120t - 3t^2) \, dt = 120 \int t \, dt - 3 \int t^2 \, dt \]
First, integrate \( 120t \):\( \int 120t \, dt = 120 \frac{t^2}{2} = 60t^2 \).
Next, integrate \( 3t^2 \): \( \int 3t^2 \, dt = 3 \frac{t^3}{3} = t^3 \).

Combine these results to form the antiderivative:
\[ P(t) = 60t^2 - t^3 + C \]
where \( C \) is the constant of integration. This constant can be determined using initial conditions, which bring us to our next core concept.

Initial conditions specify the value of the function at a particular point, helping in determining the constant of integration when solving differential equations. In our flu epidemic model, we know that at the beginning of the epidemic, the number of sick individuals is 100: \( P(0) = 100 \).

To find the constant \( C \), substitute \( t = 0 \) and \( P(0) = 100 \) into the integrated function:
\[ P(0) = 60(0)^2 - (0)^3 + C \]
This simplifies to \( 100 = C \). Therefore, \( C = 100 \).

With \( C \) known, substitute it back into the equation for \( P(t) \). The final formula for the number of people sick with the flu at time \( t \) is:
\[ P(t) = 60t^2 - t^3 + 100 \]
Thus, we've successfully modeled the flu epidemic using integration techniques and initial conditions to determine the total number of infected individuals at any time \( t \).