Question

A regional telephone company has 15 million subscribers. Each of their telephones is connected to a central office by a copper twisted pair. The average length of these twisted pairs is \(10 \mathrm{~km}\). How much is the copper in the local loops worth? Assume that the cross section of each strand is a circle \(1 \mathrm{~mm}\) in diameter, the density of copper is \(9.0 \mathrm{grams} / \mathrm{cm}^{3}\), and that copper sells for \(\$ 6\) per kilogram.

Short answer

The copper in the local loops is worth $12.717 billion.

Step-by-step solution

Calculate the Cross-sectional Area

The cross-sectional area of each strand is a circle with a diameter of 1 mm, so the radius is 0.5 mm or 0.05 cm. Use the formula for the area of a circle, \( A = \pi r^2 \), to find the area. Substitute \( r = 0.05 \) cm:\[ A = \pi \times (0.05)^2 = 0.00785 \text{ cm}^2 \].

Calculate the Volume of Copper

Each twisted pair is 10 km or 1,000,000 cm long. Use the formula for volume, \( V = A \times L \), with \( A = 0.00785 \text{ cm}^2 \) and \( L = 1,000,000 \text{ cm} \). The total volume for one loop:\[ V = 0.00785 \times 1,000,000 = 7850 \text{ cm}^3 \].

Calculate the Mass of Copper

The density of copper is 9 g/cm³. Use the density formula, \( \, \text{Density} = \frac{\text{Mass}}{\text{Volume}} \, \), rearranged to \( \, \text{Mass} = \text{Density} \times \text{Volume} \, \). Calculate the mass for one loop:\[ \text{Mass} = 9 \times 7850 = 70650 \text{ grams} = 70.65 \text{ kg} \].

Calculate the Copper Mass for All Subscribers

Each subscriber has two wires (a twisted pair), so the mass for one subscriber is double. Multiply the mass per loop by 2 to get per subscriber:\[ \text{Mass per subscriber} = 2 \times 70.65 = 141.3 \text{ kg} \].Then, multiply by the number of subscribers, 15 million:\[ \text{Total mass} = 141.3 \times 15,000,000 = 2,119,500,000 \text{ kg} \].

Calculate the Worth of the Copper

The copper sells for $6 per kg. Multiply the total mass by the price per kg to find the total worth:\[ \text{Total worth} = 2,119,500,000 \times 6 = 12,717,000,000 \text{ dollars} \].

Key concepts

Local Loops

Local loops refer to the physical connections that link a subscriber's telephone to the telephone company's central office. This connection is essential for facilitating voice and data transmission over the network. In our example, the local loop consists of a copper twisted pair, which is valued for its reliability and capacity to handle large volumes of voice traffic with minimal loss. The effective range of local loops makes them advantageous for dense urban scenarios and widespread suburban networks. Each loop's average length is 10 km, showcasing the extent of the infrastructure required to maintain connectivity for millions of subscribers.

Copper Density

Copper is a dense metal, and its density is a critical factor in calculating both the physical and economic characteristics of the local loops. With a density of 9 grams per cubic centimeter, copper is relatively heavy, enhancing its value as a conductor in electrical systems. This characteristic is pivotal in determining the mass and, subsequently, the value of the copper used in the network infrastructure. Understanding the density helps in further calculations regarding the material's cost and its efficient use in wiring and cabling applications.

Cross-Sectional Area

The cross-sectional area of a conductor, such as a copper wire, is critical in understanding its electrical properties. The problem considers a circular cross-section with a diameter of 1 mm, allowing us to calculate the area. Using the formula for the area of a circle, the radius (0.5 mm or 0.05 cm) provides an area of approximately 0.00785 square centimeters. This calculation is vital since the cross-sectional area directly affects the resistance and strength of the wire, influencing the overall efficiency of electrical transmission through the local loops.

Subscriber Lines

Subscriber lines form the backbone of telecommunications networks, linking individual users to broader communication systems. In our context, with 15 million subscribers each having a copper twisted pair, the volume and scale of material used become significant. The need for high-quality and durable materials like copper stems from the demand for consistent and uninterrupted service. Each line's interconnection with the network is a complex process requiring precision and resilience, ensuring clear communication for all subscribers.

Mass Calculation

Calculating the mass of copper required involves understanding both the volume and density of the material. With a volume computed from lengths of 10 km per subscriber loop, multiplied by two for the twisted pair configuration, the total mass integrates the calculated cross-sectional area. The mass per loop is determined to be approximately 70.65 kilograms, which is then doubled for the pair, and further multiplied by the number of subscribers (15 million). This calculation yields an astonishing total mass, reflecting the extensive amount of copper utilized across the entire network.