Problem 41 - Applied Calculus

Question

41-42. A flu epidemic hits a college community, beginning with five cases on day \(t=0\). The rate of growth of the epidemic (new cases per day) is given by the following function \(r(t),\) where \(t\) is the number of days since the epidemic began. a. Find a formula for the total number of cases of flu in the first \(t\) days. b. Use your answer to part (a) to find the total number of cases in the first 20 days. \(r(t)=18 e^{0.05 t}\)

Step-by-Step Solution

Answer

The total number of cases in the first 20 days is approximately 622.

Step 1: Understand the Problem

We are given the rate of growth of a flu epidemic, described by the function \(r(t) = 18e^{0.05t}\), where \(t\) is the number of days since the epidemic began, and \(r(t)\) is the number of new cases per day. We need to find a formula for the total number of cases in the first \(t\) days, starting with 5 cases on day 0.

Step 2: Set Up the Integral for Total Cases

The total number of cases \(C(t)\) after \(t\) days can be found by integrating \(r(t)\) from 0 to \(t\). This gives us the accumulated number of new cases over this period. Mathematically, this is expressed as:\[C(t) = 5 + \int_0^t r(u) \,du = 5 + \int_0^t 18e^{0.05u} \,du\]

Step 3: Calculate the Integral

To find \(\int_0^t 18e^{0.05u} \,du\), we will integrate the expression:\[\int 18e^{0.05u} \,du = \frac{18}{0.05}e^{0.05u} = 360e^{0.05u}.\]Thus, the definite integral from 0 to \(t\) becomes: \[\int_0^t 18e^{0.05u} \,du = 360 \,e^{0.05t} - 360.\]

Step 4: Final Formula for Total Cases

Combine the initial 5 cases and the result of the integral to form the expression for the total number of cases:\[C(t) = 5 + (360e^{0.05t} - 360) = 360e^{0.05t} - 355.\]This formula gives the total number of cases after \(t\) days.

Step 5: Calculate Total Cases for 20 Days

Now, substitute \(t = 20\) into the formula to find the total number of cases after 20 days:\[C(20) = 360e^{0.05 \times 20} - 355.\]Calculate the exponent: \(e^{1} \approx 2.718\). Then, plug it into the equation:\[C(20) = 360 \times 2.718 - 355 \approx 622.\]

Key Concepts

Calculus

Calculus is a branch of mathematics that focuses on rates of change and the accumulation of quantities. It is fundamental in understanding how things change over time, which is essential in modeling real-world phenomena like flu epidemics. Calculus helps us study the behavior of functions and allows us to analyze changes, such as the rate at which the flu spreads.
One of the key components of calculus is differentiation, which deals with rates of change. In the original problem, the rate of new flu cases is modeled by the function \(r(t) = 18e^{0.05t}\). This represents the derivative of the total number of cases with respect to time \(t\).
Another core part of calculus is integration, which is essentially the opposite of differentiation. It revolves around the accumulation of quantities, such as figuring out the total number of flu cases by integrating the rate function \(r(t)\) over time. This provides a more concrete understanding of how calculus is used in modeling and solving dynamic real-life problems by moving from known rates to total amounts.

Exponential Growth

Exponential growth is when a quantity increases at a rate proportional to its current value. This pattern is commonly observed in natural phenomena like populations, investments, or epidemics.
In the context of the flu epidemic problem, the rate of new cases \(r(t) = 18e^{0.05t}\) is an exponential function. This means that the number of new cases each day grows exponentially, becoming faster as times goes on. Essentially, every day the increase in cases is a percentage of the total that reached a certain level the previous day.
Understanding this concept is crucial because it shows how quickly an epidemic can spread if uncontrolled. Initially, there may be just a few cases, but being exponential, the number grows rapidly and continues to do so as long as conditions remain favorable for the spread. Recognizing this growth can help in planning interventions to effectively manage or mitigate the spread of disease.

Integration

Integration is a foundational concept in calculus that allows us to find the accumulated value of quantities. In practical terms, when you know the rate of change of something, integration lets you determine the total amount over a period.
In the exercise, we start with the rate of new flu cases as given by \(r(t) = 18e^{0.05t}\). To find the total number of cases over the first \(t\) days, we integrate this function. The integral sums up all the little changes over time, giving us the accumulated total.
Mathematically, this is expressed through an integral such as \(\int_0^t 18e^{0.05u} \,du\). By performing this operation, we derive the formula for the total number of cases, \(C(t) = 360e^{0.05t} - 355\).
This highlights how integration converts rates into totals, making it an invaluable tool in analyzing phenomena like epidemic modeling. Understanding integration is crucial as it connects instantaneous changes to cumulative outcomes, providing insights into how quantities build over time.