Problem 35 - 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 m long and 0.005 m in diameter has a resistance of 140 ohms. (c) Find the resistance of a wire made of the same material that is 3 m long and has a diameter of 0.008 m.

Step-by-Step Solution

Answer

(a) \( R = k \frac{L}{d^2} \); (b) \( k \approx 0.0014583 \); (c) \( R \approx 68.27 \) ohms.

Step 1: Understanding Joint Variation

In a joint variation problem, a variable depends on multiple other variables. Here, the resistance \( R \) varies directly with the length \( L \) and inversely with the square of the diameter \( d \). The equation for this relationship can be written as \( R = k \frac{L}{d^2} \), where \( k \) is the constant of proportionality.

Step 2: Solving for Constant of Proportionality

Using the given information, substitute \( L = 1.2 \) m, \( d = 0.005 \) m, and \( R = 140 \) ohms to find \( k \). Plug these into the equation: \( 140 = k \frac{1.2}{(0.005)^2} \). After simplifying, we solve for \( k \): \[ k = \frac{140 \times 0.005^2}{1.2} = 0.0014583 \] (approximately).

Step 3: Calculating the New Resistance

With the constant of proportionality \( k \) found, substitute \( L = 3 \) m and \( d = 0.008 \) m into the equation to find the new resistance. Using \( R = 0.0014583 \frac{3}{(0.008)^2} \), we compute : \[ R = 0.0014583 \frac{3}{0.000064} = 68.27 \] ohms (approximately).

Key Concepts

Constant of Proportionality

The constant of proportionality, often symbolized as \( k \), is a crucial value in equations involving direct and inverse variation. It serves as a multiplier that allows the relationship between different variables to maintain consistency. In problems involving joint variation, such as the resistance of a wire, the constant of proportionality helps define how one variable changes in relation to another. For instance, when we say resistance \( R \) varies directly with length \( L \) and inversely with the square of the diameter \( d \), the equation is \( R = k \frac{L}{d^2} \). Here, \( k \) ensures that any change in \( L \) or \( d \) results in a predictable change in \( R \).
To find \( k \), we often use given values for the variables. As seen in the original exercise, by substituting \( R = 140 \), \( L = 1.2 \), and \( d = 0.005 \) into the equation, we solve for \( k \), finding it to be approximately 0.0014583.

Electrical Resistance

Electrical resistance is a measure of how difficult it is for electric current to pass through a conductor. It is heavily affected by various physical properties of the wire, such as its length and diameter. Longer wires typically have greater resistance since the current must travel further, whereas thicker wires (with a larger diameter) offer less resistance.
The relationship investigated in the exercise demonstrates how resistance \( R \) depends directly on length \( L \) and inversely on the square of diameter \( d \). This means that as the wire gets longer, its resistance increases, while a bigger diameter will decrease resistance. These principles are crucial in designing electrical circuits and components, ensuring the correct materials and dimensions are chosen for optimal functionality.

Direct Variation

Direct variation describes a relationship where one variable increases as another increases. In the equation \( R = k \frac{L}{d^2} \), the resistance \( R \) has a direct variation with the length \( L \) of the wire. This implies that if all other factors remain constant, doubling the length of the wire will double the resistance.
Understanding direct variation helps in predicting how changing one aspect of a situation influences others. For example, in an electric circuit, if the aim is to intentionally increase resistance, increasing the length of the wire can be a straightforward approach. Direct variations are linear relationships and are relatively simple to compute and understand.

Inverse Variation

Inverse variation occurs when one variable increases while another decreases. In our given equation \( R = k \frac{L}{d^2} \), the resistance \( R \) inversely varies with the square of the diameter \( d \). This implies that as the diameter of the wire increases, the resistance decreases.
Inverse variation is an essential concept in fields like physics and engineering. For the resistance of a wire, if you were to double the diameter, the resistance would be reduced to a quarter of its previous value, assuming the length remains the same. Recognizing the effects of inverse variation helps in optimizing materials for desirable electrical properties. Understanding these relationships is key to controlling and manipulating electrical systems effectively.