Problem 86 - Calculus and its applications

Question

Healing wound. The circular area \(A\), in square centimeters, of a healing wound is approximated by $$ A(r)=3.14 r^{2} $$ where \(r\) is the wound's radius, in centimeters. a) Find the rate of change of the area with respect to the radius. b) Find \(A^{\prime}(3)\). c) Explain the meaning of your answer to part (b).

Step-by-Step Solution

Answer

The area increases at 18.84 cm²/cm when the radius is 3 cm.

Step 1: Understand the Formula

We are given the formula for the area of a circular wound, which is defined as \( A(r) = 3.14r^2 \). Here, \( A(r) \) is the area, and \( r \) is the radius of the wound.

Step 2: Find the Rate of Change of the Area

To find the rate of change of the area with respect to the radius, we need to differentiate the area function \( A(r) \) with respect to \( r \). We use the power rule for differentiation: if \( f(r) = ar^n \), then \( f'(r) = nar^{n-1} \). Thus, the derivative of \( A(r) = 3.14r^2 \) with respect to \( r \) is \( A'(r) = 2 \times 3.14 \times r = 6.28r \).

Step 3: Calculate \(A'(3)\)

Now, substitute \( r = 3 \) into \( A'(r) = 6.28r \) to find \( A'(3) \). Calculate \( A'(3) = 6.28 \times 3 = 18.84 \).

Step 4: Interpret \(A'(3)\)

\( A'(3) = 18.84 \) means that when the radius of the wound is 3 cm, the area of the wound is increasing at a rate of 18.84 square centimeters per centimeter with respect to the radius.

Key Concepts

Differentiation

Differentiation is a fundamental concept in calculus. It's all about finding how a function changes as its input changes. Imagine you want to know how fast something grows or shrinks at a specific point. That's where differentiation comes in! By calculating a derivative, we can find the rate at which one quantity changes concerning another.
For example, in the problem of the healing wound, we're interested in the change in area as the radius changes. This can help in understanding how quickly the wound is healing as it grows. Differentiation involves rules, such as the power rule, to help compute these rates of change efficiently.
Understanding differentiation is crucial as it provides insights into the behavior of functions and can be applied in various fields like physics, engineering, and economics.

Power Rule in Calculus

The power rule is one of the simplest and most commonly used tools in differentiation. It helps us to find the derivative of functions that are in the form of a power. The rule states:

• If you have a function of the form: \( f(r) = ar^n \),
• Its derivative is: \( f'(r) = nar^{n-1} \).

To use the power rule, you simply bring down the power as a coefficient and then reduce the power by one. In our exercise, the function \( A(r) = 3.14r^2 \) was differentiated using this rule. The power of 2 was multiplied by the coefficient 3.14, resulting in \( A'(r) = 6.28r \).
This rule allows us to quickly find rates of change without needing to conduct more complicated calculations. It's a handy shortcut that simplifies working with polynomial functions.

Derivatives Interpretation

Once we find a derivative, it's important to interpret what it means. Derivatives can tell us how fast a quantity is changing. For the healing wound case, interpreting the derivative \( A'(3) = 18.84 \) is important because it explains how the area of the wound is increasing as the radius expands.
Here's what it means:

• At the exact moment when the radius is 3 cm, the area is growing by 18.84 square centimeters for every 1 cm increase in the radius.
• This interpretation can be valuable for medical professionals to understand the growth rate of a healing wound.

Understanding interpretation helps in drawing real-world conclusions from mathematical results. It's not just about getting to a number; it's about understanding the significance of the number in the original context of the problem.