Question

Jewelry A jeweler resizes a ring so that its inner circumference is 6 centimeters. (a) What is the radius of the ring? (b) The inner circumference of the ring varies between 5.5 centimeters and 6.5 centimeters. How does the radius vary? (c) Use the \(\varepsilon-\delta\) definition of limit to describe this situation. Identify \(\varepsilon\) and \(\delta .\)

Short answer

The radius of the ring is \(3/\pi\) cm when the inner circumference is 6 cm. The radius varies between \(2.75/\pi\) cm and \(3.25/\pi\) cm when the inner circumference varies between 5.5 cm and 6.5 cm. Using the epsilon-delta definition, \(epsilon\) is \(0.5\) cm and \(delta\) is \((0.5)/(2\pi)\) cm.

Step-by-step solution

Find the radius

Use the formula for the circumference of a circle, \(C = 2\pi r\) where \(C\) is the circumference and \(r\) is the radius. The task is to solve for the radius of the ring, which whose inner circumference is given as \(6\) centimeters. Rearrange the formula to solve for \(r\): \(r = C/(2\pi)\). Substitute \(C = 6\) into the rearranged formula: \(r = 6/(2\pi) = 3/\pi\)

Calculate the variability of the radius

The inner circumference of the ring varies between 5.5 cm and 6.5 cm. Repeat the process from Step 1 for \(C = 5.5\) and \(C = 6.5\). The calculations are: \(r_1 = 5.5/(2\pi) = 2.75/\pi\), \(r_2 = 6.5/(2\pi) = 3.25/\pi\). The variability is the difference between these values: \((3.25-2.75)/\pi\)

Apply the epsilon-delta definition

In this case, \(epsilon\) is used to represent the variability in the circumference, which is \(0.5\) cm. \(delta\) is used to represent the corresponding variability in the radius, which is \((0.5)/(2\pi)\) cm. The epsilon-delta definition of limit expresses the idea that for any amount of variability \(epsilon\), however small, there is some corresponding variability \(delta\) such that if the radius changes by less than \(delta\), the circumference will change by less than \(epsilon\).

Key concepts

Circumference of a Circle

Understanding the circumference of a circle is fundamental in geometry and various real-world applications like sizing jewelry. The circumference refers to the distance around the edge of a circle, akin to the perimeter of polygons. It can be calculated using the formula \( C = 2\pi r \) where \( C \) stands for circumference and \( r \) represents the circle's radius. In simple terms, if we were to unwrap a circle and measure the length of the edge, that measurement would be the circumference.

For instance, when a jeweler needs to resize a ring, using the circumference is key to ensuring a perfect fit. If we’re told that a ring's inner circumference is 6 centimeters, the formula provides a straightforward way to determine the exact size of the ring in terms of its radius—crucial information for resizing efforts.

Radius Calculation

The radius of a circle is the distance from the center to any point on its edge. Calculating the radius is often a necessity in tasks such as determining the size of a ring for jewelry design. Once we have the circumference (\( C \) cm), we can use the rearranged formula \( r = \frac{C}{2\pi} \) to find the radius. By substituting the given circumference into this formula, we arrive at the ring's radius.

For the exercise at hand, with a circumference of 6 cm, the formula yields a radius of \( r = \frac{6}{2\pi} = \frac{3}{\pi} \) cm. It is important for students to grasp this process as it illustrates a direct application of math concepts to practical problems such as jewelry design.

Variability in Mathematics

Variability is a concept used to describe the extent of change or difference within a set of values. In mathematics, it is essential to quantify and manage variability to predict and adjust for possible outcomes. In the context of our exercise, the inner circumference of a ring can vary, indicating that the size is not fixed but fluctuates within a range, specifically between 5.5 cm and 6.5 cm.

When this fluctuation is translated to the radius of the ring, we operate similar calculations as before to determine the radius for both the minimum and maximum circumferences. This gives us an understanding of the lower and upper bounds for the ring's radius, which visually represents the concept of variability in a practical scenario.

Concept of Limits

The concept of limits is a cornerstone of calculus, serving to define exact values as a function approaches a certain point. In simpler terms, it calculates the value that a variable gets infinitely close to as the conditions change. This is critical in understanding continuous changes in variables, and is expressed precisely by the epsilon-delta (\(\varepsilon-\delta\))) definition of limit.

The \(\varepsilon-\delta\))) approach allows us to quantitatively describe how the variability in one value (like the circumference of a ring) corresponds to variability in another (like the ring's radius). In the exercise, we identify \(\varepsilon\))) as the change in circumference and \(\delta\))) as the consequent change in radius. This mathematical framework provides a rigorous way to predict and discuss small changes within systems, highlighting the meticulous nature of the mathematics that underlie even the most mundane situations, such as sizing a piece of jewelry.