Problem 41 - Algebra and Trigonometry

Question

Electrical Resistance The resistance \(R\) of a wire varies directly as its length \(L\) and inversely as the square of its diameter \(d\). (a) Write an equation that expresses this joint variation. (b) Find the constant of proportionality if a wire 1.2 \(\mathrm{m}\) long and 0.005 \(\mathrm{m}\) in diameter has a resistance of 140 ohms. (c) Find the resistance of a wire made of the same material that is 3 \(\mathrm{m}\) long and has a diameter of \(0.008 \mathrm{m} .\)

Step-by-Step Solution

Answer

The resistance equation is \( R = k \frac{L}{d^2} \), where \( k = 0.00291667 \). The new wire's resistance is 136.41 ohms.

Step 1: Understanding Direct and Inverse Variation

The resistance \(R\) varies directly as the length \(L\) and inversely as the square of the diameter \(d\). If \( R \) is directly proportional to \( L \), we have \( R \propto L \). Similarly, if \( R \) is inversely proportional to \( d^2 \), we have \( R \propto \frac{1}{d^2} \). Combining these, we get \( R \propto \frac{L}{d^2} \).

Step 2: Writing the Equation

To express the joint variation as an equation, introduce a constant of proportionality \( k \). Therefore, the equation becomes \( R = k \frac{L}{d^2} \).

Step 3: Calculate the Constant of Proportionality

Using the given values: \( L = 1.2 \) m, \( d = 0.005 \) m, and \( R = 140 \) ohms, plug these into the equation: \( 140 = k \frac{1.2}{(0.005)^2} \). Calculate \( (0.005)^2 = 0.000025 \), then \( \frac{1.2}{0.000025} = 48000 \). Thus, \( k = \frac{140}{48000} = 0.00291667 \).

Step 4: Calculate the Resistance of a New Wire

For a new wire, \( L = 3 \) m and \( d = 0.008 \) m, use the constant \( k = 0.00291667 \). Plug these values into the equation \( R = k \frac{L}{d^2} \): \( d^2 = 0.008^2 = 0.000064 \). Compute \( \frac{3}{0.000064} = 46875 \) and then \( R = 0.00291667 \times 46875 = 136.41 \) ohms.

Key Concepts

Direct Variation

Direct variation occurs when two quantities increase or decrease together at a constant rate.
In the case of electrical resistance, we say that the resistance \( R \) of a wire is directly proportional to its length \( L \). This is because as the length of the wire increases, the resistance also increases, assuming other factors remain constant.

• The concept is modeled using the equation \( R \propto L \).
• In simpler terms, if you double the length of the wire, the resistance will also double, keeping diameter constant.

Direct variation is a fundamental concept in understanding many physical relationships, illustrating how change in one variable leads to a predictable change in another.

Inverse Variation

Inverse variation describes a relationship where one quantity increases while the other decreases.
For electrical resistance, this means that the resistance \( R \) varies inversely with the square of the wire's diameter \( d \).

• Mathematically, this is expressed as \( R \propto \frac{1}{d^2} \).
• This implies that as the diameter is increased, the resistance decreases.

To further elaborate, when the diameter of the wire is made larger, current can flow more freely, thus decreasing resistance. In practical terms, thinner wires tend to have higher resistance because electric charges have less space to move through.

Constant of Proportionality

The constant of proportionality \( k \) helps to quantify direct and inverse variations. It provides a precise mathematical link between variables.
For the equation \( R = k \frac{L}{d^2} \), \( k \) integrates all other material-specific factors influencing resistance.

• This constant is achieved by using known values for \( R \), \( L \), and \( d \).
• In our case, given \( R = 140 \) ohms, \( L = 1.2 \) m, and \( d = 0.005 \) m, we calculated \( k \) to be approximately \( 0.00291667 \).

Understanding the constant of proportionality allows us to predict resistance for different lengths and diameters of the same material.

Joint Variation

Joint variation combines both direct and inverse variation principles into a single relationship. This is used when a variable depends on multiple other variables.
In the context of electrical resistance, joint variation explains how resistance \( R \) depends on both the length of the wire \( L \) and the square of the diameter \( d\).

• The equation \( R = k \frac{L}{d^2} \) demonstrates this dual dependency.
• The resistance varies directly with length while inversely with the square of the diameter.

This type of variation is common in physical laws, allowing for complex relationships to be represented in a simple mathematical form. By understanding joint variation, we can make accurate predictions about how changing multiple factors influences others.