Question

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

Short answer

The radius of the ring, when the inner circumference is 6cm, is approximately 0.955cm. If the ring's inner circumference can vary between 5.5cm and 6.5cm, then the radius will vary between some 0.875cm and 1.034cm. The situation can be represented through the \(\varepsilon-\delta\) definition of a limit, with \(\varepsilon\) representing the change in the ring's inner circumference, and \(\delta\) representing the change in the ring's radius. In this context, \(\varepsilon=|C-6|\) and \(\delta=|r-0.955|\).

Step-by-step solution

Calculate the radius of ring

To calculate the radius of the ring, you make use of the formula \(C = 2\pi r\), where \(C\) is the circumference and \(r\) is the radius. Rearrange the formula to solve for the radius \(r\) to give \(r = C /(2 \pi)\). Plug in the given value for \(C\) (6cm) to get the radius \(r = 6 / (2 \pi) \approx 0.955 cm\).

Calculate the range of radius

Given that the ring's inner circumference can vary between 5.5 cm and 6.5 cm, the radius will correspondingly vary between \(r_{min} = 5.5 / (2 \pi)\) and \(r_{max} = 6.5 / (2 \pi)\). Calculating these values will give \(r_{min} \approx 0.875 cm\) and \(r_{max} \approx 1.034 cm\).

Define \(\varepsilon-\delta\) in this context

In the \(\varepsilon-\delta\) scenario, the variation in the ring's inner circumference can be represented by \(\varepsilon\) and the variation in the ring's radius can be represented by \(\delta\). Given that \(\varepsilon\) represents the change in the function's output and \(\delta\) represents the change in the function's input, in this case, \(\varepsilon=|C-6|\) and \(\delta=|r-0.955|\) where \(C\) represents the output of the function (the circumference) and \(r\) the input to the function (the radius).

Key concepts

Circumference and Radius in Jewelry Design

Understanding the relationship between the circumference and radius of a circle is fundamental in designing jewelry such as rings. The circumference refers to the total length around the circle, which in the case of a ring translates to the distance around the inner section of the band. To find the radius—a term that denotes the straight-line distance from the center of the circle to any point on its perimeter—you can use the formula \( C = 2\pi r \).

In practical terms, if a jeweler needs a ring with an inner circumference of 6 centimeters, applying the formula \( r = \frac{C}{2\pi} \) allows for the calculation of the radius. For a circumference of 6 cm, the radius is approximately 0.955 cm. This calculation is critical because it impacts how the ring will fit on a finger, which is vital for both aesthetics and comfort.

For jewelers, being thorough with these calculations ensures that the designs they conceive can be brought to life accurately, thereby providing clients with well-fitting jewelry.

• Circumference: The full length around the circle (the ring's band)
• Radius: The distance from the center to any point on the edge (half of the ring's diameter)

Epsilon-Delta Definition of a Limit in Jewelry Sizing

In calculus, the ε-δ (epsilon-delta) definition of a limit describes the precision of limits using two variables: ε (epsilon) for output variations and δ (delta) for input variations. This mathematical concept may seem abstract but can be quite useful in practical applications such as jewelry design.

When resizing a ring, a jeweler might define an acceptable range for the inner circumference. This allowable variation can be described using the ε-δ language. Epsilon represents the maximum allowable deviation of the circumference from its intended value, and delta represents the corresponding change in the radius. For instance, if a ring is meant to have a 6 cm circumference, but it can vary between 5.5 and 6.5 cm, then ε would be 0.5 cm.

Using the ε-δ definition helps jewelers to communicate and quantify the precision needed in their work, making the transition from theoretical mathematics to hands-on craftsmanship. This ensures that even subtle changes in measurements can be clearly understood and accounted for during the design and fabrication process.

• Epsilon (ε): Acceptable output variation (circumference)
• Delta (δ): Corresponding input variation (radius)

Managing Variability in Measurements

Variability in measurements is an inevitable part of any design process, including in delicate fields like jewelry making. When a jeweler states that a ring's inner circumference can range from 5.5 to 6.5 centimeters, this indicates a variability or tolerance in the final product. It’s essential to calculate how this affects the radius to ensure the ring fits properly.

By calculating the minimum and maximum radius—0.875 cm and 1.034 cm, respectively, in our example—the jeweler can understand and control the variability in ring sizes. An accurate assessment of this variability is crucial for meeting customers’ expectations and for maintaining quality control in jewelry production.

Ultimately, recognizing and accommodating measurement variability allows jewelers to set realistic expectations and ensure customer satisfaction. Well-defined tolerances also prevent costly remakes and adjustments, sustaining both the art and the business of jewelry design.

• Variability in Measurements: The range within which the actual measurement can fluctuate
• Quality Control: Ensuring the ring fits within the specified variability range
• Customer Satisfaction: Providing a product that meets the expected size and fit