Problem 32 - Calculus for Biology and Medicine

Question

Suppose that a fungal disease originates in the middle of an orchard, initially affecting only one tree. The disease spreads out radially at a constant speed of 10 feet per day. What area will be affected after 2 days, 4 days, and 8 days? Write an equation that expresses the affected area as a function of time, measured in days, and show that this function is a polynomial of degree \(2 .\)

Step-by-Step Solution

Answer

The affected area after 2, 4, and 8 days is \( 400\pi \), \( 1600\pi \), and \( 6400\pi \) square feet, respectively.

Step 1: Understanding the Problem

We are told that a fungal disease starts in one tree and spreads radially at a constant speed of 10 feet per day. We need to determine the affected area after 2, 4, and 8 days, and express the affected area as a function of time. Furthermore, we will verify that this function is a polynomial of degree 2.

Step 2: Determine the Radius

The disease spreads at 10 feet per day. Thus, the radius of the affected area is the speed multiplied by time. If we let \( t \) represent time in days, then the radius \( r \) is given by the equation \( r = 10t \).

Step 3: Calculate the Affected Area

The affected area can be represented as a circle with a radius \( r = 10t \). The area of a circle is \( \pi r^2 \). Substituting for the radius, the affected area \( A \) is given by \( A = \pi (10t)^2 = 100\pi t^2 \).

Step 4: Verify the Function's Degree

The function for the affected area is \( A(t) = 100\pi t^2 \). This expression is a polynomial because it can be written in the form \( a_n t^n + a_{n-1} t^{n-1} + ... + a_0 \). Here, the highest power of \( t \) is 2, confirming that it is a polynomial of degree 2.

Step 5: Compute Area for Specific Days

Now we use \( A(t) = 100\pi t^2 \) to find the area after 2, 4, and 8 days. - After 2 days: \( A(2) = 100\pi (2)^2 = 400\pi \) square feet. - After 4 days: \( A(4) = 100\pi (4)^2 = 1600\pi \) square feet. - After 8 days: \( A(8) = 100\pi (8)^2 = 6400\pi \) square feet.

Key Concepts

Radial Growth

When a disease, such as a fungal infection, spreads radially, it means it expands outward from a single point in all directions. In the context of the exercise, this disease begins at one tree and spreads like ripples on a pond. The key aspect of radial growth here is that the disease moves outward at a steady rate, in this case, 10 feet per day.

Understanding radial growth is important because it allows us to determine how far the disease will spread over time. The radius, which is the distance from the center of the initial infection to the edge of the affected area, increases linearly with time. This means you can calculate the radius at any given time using a simple formula.

• Formula for the radius: \( r = \text{speed} \times \text{time} \)
• For this problem: \( r = 10t \) (where \( t \) is in days)

Knowing this can help us calculate not just how far the infection spreads, but also the total area affected, which is the next step in understanding this problem.

Affected Area Calculation

Once you have the radial growth, determining the affected area becomes straightforward. Because the disease spreads outwards in a circle, the area affected can be calculated using the formula for the area of a circle, which requires the radius.

In mathematical terms, the area \( A \) of a circle is given by the formula:

• \( A = \pi r^2 \)

Given that we have earlier established the radius as \( r = 10t \), this is plugged into the area formula to express the area in terms of time, \( t \):

• \( A = \pi (10t)^2 \)
• \( A = 100\pi t^2 \)

This formula tells us the affected area as a function of time, providing a clear picture of how the spread increases over days. Moreover, we can quickly determine the affected area on specific days by plugging values of \( t \) into the formula. For instance:

• \( A(2) = 400\pi \) square feet
• \( A(4) = 1600\pi \) square feet
• \( A(8) = 6400\pi \) square feet

This method of using a mathematical formula gives a precise representation of how an infectious disease's impact area grows over time.

Degree of Polynomial

In mathematics, a polynomial is an expression that consists of variables and coefficients, combined using only the operations of addition, subtraction, multiplication, and non-negative integer exponents. The degree of a polynomial is the highest degree of its terms when the polynomial is expressed in its standard form.

The function defined for the affected area is \( A(t) = 100\pi t^2 \). This expression is identified as a polynomial because it is structured as \( a_n t^n + a_{n-1} t^{n-1} + \ldots + a_0 \), which fits the definition of a polynomial with a single term.

• The term \( 100\pi t^2 \) has a degree of 2 since the variable \( t \) is raised to the 2nd power.

Therefore, the affected area function is a quadratic polynomial, confirming its degree as 2. Recognizing the degree of a polynomial is essential because it highlights how the output (in this case, the affected area) grows with changes in the input (days of spreading). The degree indicates that the relationship between time and affected area is not linear but quadratic; thus, the area grows faster over time than it would with a first-degree relation. Understanding these mathematical expressions helps to predict and manage the spread of phenomena like disease outbreaks.