Problem 45 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Cathy will fence off a circular pen for her rabbits. Express the area of the rabbit pen as a function of the length of fencing she uses.

Step-by-Step Solution

Answer

The area of the rabbit pen as a function of the length of fencing used is \(A = (C^2) / (4\pi)\).

Step 1: Obtain Radius from Fencing Used

First, express the radius of the circular pen in terms of the length of fencing used. To do this, resolve the formula \(C = 2\pi r\) for the radius by isolating \(r\). Therefore, divide the total length of fencing \(C\) by \(2\pi\). This gives us the radius as \(r = C / (2\pi)\). Note that \(C\) is the total length of fencing used.

Step 2: Substitution Into Area Formula

Substitute the radius \(r = C / (2\pi)\) obtained in Step 1 into the area formula \(A = \pi r^2\). This allows us to express the area in terms of the length of fencing \(C\). So we get \(A = \pi [C / (2\pi)]^2\).

Step 3: Simplify the Expression

Simplify the equation obtained in Step 2. By doing so we get \(A = (C^2) / (4\pi)\). This final formula represents the area of the circular pen as a function of the length of the fencing she uses.

Key Concepts

Fencing and Area Relationship

Understanding the relationship between fencing and the area of a circular pen is essential in optimizing the use of materials while ensuring sufficient space for pets like Cathy's rabbits. When Cathy builds a circular pen for her rabbits, the length of fencing required will directly determine the size of the area available to the rabbits.

To clarify, the fencing outlines the circumference of the circle, which mathematically is the total length around it. The formula to calculate the area of a circle is expressed as \(A = \text{pi} r^2\), where \(A\) represents the area, and \(r\) is the radius of the circle. To find this area using the length of fencing, which is essentially the circumference \(C\), we must first find the relation between \(C\) and \(r\). Once we have \(r\), we can use it to compute the area that will be enclosed by the fencing.

This process effectively turns the practical, physical act of fencing into a mathematical model which can assist Cathy in understanding how much space her rabbits will have to hop around.

Circumference to Radius

The transition from the circumference to the radius is a pivotal step in determining the pen area from the length of fencing. The circumference, which we denote as \(C\), is the total length of the fencing that Cathy will use to create the pen. Using the formula \(C = 2\text{pi} r\), we can isolate the radius \(r\) by dividing the circumference by \(2\text{pi}\).

This mathematical manipulation gives us the unique radius for any given circle when we only know the perimeter length, such as our case with the fencing. Specifically, the formula becomes \(r = \frac{C}{2\text{pi}}\). This formula is essential because it allows us to convert the linear measurement of the fence into the radial dimension of the pen, which forms the base for calculating the area.

Grasping this concept is critical for students as it showcases an everyday practical application of geometry, transforming a physical length of fencing into a crucial calculable value for area determination.

Simplifying Mathematical Expressions

Once the substitution of the radius is made into the area formula, we arrive at a more complex mathematical expression which needs to be simplified. The simplification process is vital because it leads to a clearer understanding of the relationship between variables and makes computations easier.

In our scenario, after substituting the derived radius into the area formula, we end up with the equation \(A = \pi \left(\frac{C}{2\pi}\right)^2\). Simplifying this expression involves distributing the exponent and canceling out common terms. Here, the \(\pi\) in the numerator and one of the \(\pi\)s in the denominator cancel out, and the exponent of 2 distributes to both numerator and denominator of the fraction separately, ultimately yielding the much neater expression \(A = \frac{C^2}{4\pi}\).

Simplification in mathematics is an invaluable skill that enables students to deconstruct complex problems into more manageable segments. This can lead to better problem-solving abilities and a deeper comprehension of underlying mathematical principles.