Problem 37 - Intermediate Algebra

Question

The electrical resistance of a wire varies directly as its length and inversely as the square of its diameter. If the resistance of 200 meters of wire that has a diameter of \(\frac{1}{2}\) centimeter is \(1.5\) ohms, find the resistance of 400 meters of wire with a diameter of \(\frac{1}{4}\) centimeter.

Step-by-Step Solution

Answer

The resistance of the new wire is 12 ohms.

Step 1: Understand the Relationship

The resistance \( R \) of a wire varies directly with its length \( L \) and inversely with the square of its diameter \( d \). This can be expressed with the equation \( R = k \cdot \frac{L}{d^2} \), where \( k \) is a constant of proportionality.

Step 2: Find the Constant k

Given that for a wire of length \( L = 200 \) meters and diameter \( d = \frac{1}{2} \) cm, the resistance \( R = 1.5 \) ohms, substitute these values into the equation: \( 1.5 = k \cdot \frac{200}{(\frac{1}{2})^2} \). This simplifies to \( 1.5 = k \cdot \frac{200}{\frac{1}{4}} \). Solve for \( k \) by calculating \( k = 1.5 \times \frac{1}{4} \times \frac{1}{200} = 0.001875 \).

Step 3: Set Up the Equation for New Wire

Now, we need to find the resistance for the new wire with \( L = 400 \) meters and \( d = \frac{1}{4} \) cm. Use the equation \( R = k \cdot \frac{L}{d^2} \) with \( k = 0.001875 \). Substitute the values: \( R = 0.001875 \cdot \frac{400}{(\frac{1}{4})^2} \).

Step 4: Calculate the Resistance

Simplify the expression: \( R = 0.001875 \cdot \frac{400}{\frac{1}{16}} = 0.001875 \times 6400 = 12 \). Therefore, the resistance of the second wire is \( 12 \) ohms.

Key Concepts

Direct and Inverse Variation

Understanding direct and inverse variation is crucial to grasp the concept of electrical resistance in wires. When something varies directly with another quantity, it means that as one quantity increases, the other also increases, and vice versa. In our case, the resistance \( R \) of a wire varies directly with its length \( L \). This implies that if the length of the wire increases, the resistance increases proportionally. For example, doubling the length of the wire would theoretically double its resistance, assuming all other factors remain constant.

Conversely, inverse variation is when one quantity increases and the other decreases. Here, resistance varies inversely with the square of the wire's diameter \( d \). This means if the diameter of the wire doubles, the resistance does not just halve, but becomes one-fourth of its original value because the relationship involves the square of the diameter \((d^2)\). This relationship can be defined by an equation:

• \( R = k \cdot \frac{L}{d^2} \)
• Where \( k \) is a constant

The equation shows that resistance is directly proportional to length and inversely proportional to the square of diameter.

Proportionality Constant

A proportionality constant \( k \) is a constant value used to relate quantities that are directly or inversely proportional in mathematical equations. In the context of electrical resistance, \( k \) helps us understand how the various properties of the wire – length and diameter – interact.

To find \( k \), we use given conditions. For instance, from the original exercise, we were provided with a scenario where the resistance \( R = 1.5 \) ohms for a wire of length \( 200 \) meters and diameter \( \frac{1}{2} \) cm. By substituting these values into the equation \( R = k \cdot \frac{L}{d^2} \), you can solve for \( k \):

• \( 1.5 = k \cdot \frac{200}{(\frac{1}{2})^2} \)
• Solve for \( k \) to find that \( k = 0.001875 \)

Having found \( k \), you can then use this constant when calculating the resistance of other wire configurations, making it an essential part of understanding and solving real-world problems related to electrical resistance.

Wire Length and Diameter

The wire's length and diameter have pivotal roles in determining its electrical resistance. The length \( L \) of a wire affects its resistance because the electrons traveling through the wire have more material to pass through, leading to more collisions and therefore higher resistance as the distance increases.

On the other hand, the diameter \( d \) affects the resistance inversely, and significantly so, due to the square term in the variation equation. A larger diameter reduces resistance because there's more space for electric current to pass through, resulting in fewer collisions among electrons. Essentially, a thicker wire provides an easier path for current flow as compared to a thinner one.

When solving problems or experimenting, it's important to remember:

• More length means more resistance
• More diameter means much less resistance

These principles are essential not only for academic exercises but also for practical applications in designing efficient electrical circuits where resistance impacts the performance and safety of electrical systems.