Problem 41 - Precalculus Mathematics for Calculus

Question

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

(a) Equation: \( R = k \frac{L}{d^2} \). (b) \( k = 0.0029167 \). (c) Resistance is approximately 136.375 ohms.

Step 1: Understanding Direct and Inverse Variation

When we say the resistance \( R \) varies directly with length \( L \), it means \( R \) increases as \( L \) increases. When varying inversely with \( d^2 \) (the square of its diameter), \( R \) decreases as \( d \) increases. Hence, we can express this relationship as \( R = k \frac{L}{d^2} \), where \( k \) is the constant of proportionality.

Step 2: Substituting Known Values to Find k

We substitute the given values into our equation to find \( k \). With \( L = 1.2 \) meters, \( d = 0.005 \) meters, and \( R = 140 \) ohms, the equation becomes \( 140 = k \frac{1.2}{(0.005)^2} \). Simplify to find \( k \).

Step 3: Calculating k

Calculate \( (0.005)^2 = 0.000025 \), then the equation becomes \( 140 = k \frac{1.2}{0.000025} \). Solve for \( k \): \( k = \frac{140 \times 0.000025}{1.2} = 0.0029167 \).

Step 4: Setting Up Equation for New Wire

Use the equation \( R = 0.0029167 \frac{L}{d^2} \) with the new wire specifications \( L = 3 \) meters, \( d = 0.008 \) meters.

Step 5: Calculating Resistance for New Wire

Substitute \( L = 3 \) meters and \( d = 0.008 \) meters into the equation: \( R = 0.0029167 \times \frac{3}{(0.008)^2} \). First, calculate \( (0.008)^2 = 0.000064 \). Then, \( R = 0.0029167 \times \frac{3}{0.000064} \).

Step 6: Final Resistance Calculation

Calculate \( R = 0.0029167 \times 46875 = 136.375 \). Thus, the resistance of the new wire is approximately 136.375 ohms.

Key Concepts

Resistance Calculation

When dealing with the resistance of a wire, understanding how different attributes of the wire affect resistance is key. Resistance (R) in a wire is influenced by two main factors: its length (L) and its diameter (d). To calculate this resistance, the following principles are applied:

• Resistance varies directly with length: This means if the length of the wire increases, the resistance would also increase.
• Resistance varies inversely with the square of the diameter: Contrary to length, if the diameter increases, the resistance decreases because it's inversely related to the square of the diameter (d^2).

Combining these variations, the resistance equation forms as: \[ R = k \frac{L}{d^2} \]where k is the proportionality constant. This formula helps determine the resistance when we know the length and diameter of a wire.

Proportionality Constant

The proportionality constant (k) is a crucial part of the resistance calculation equation. This constant bridges the direct and inverse relationship between length, diameter, and resistance in a wire. To find k, you use specific known values of resistance, length, and diameter. For instance:

• Take the formula \[ R = k \frac{L}{d^2} \]
• Insert the known values \( R = 140 \) ohms, \( L = 1.2 \) meters, and \( d = 0.005 \) meters to solve for k.
• The equation becomes \[ 140 = k \frac{1.2}{(0.005)^2} \], which simplifies to finding \( k = 0.0029167 \).

The value of k enables us to calculate resistance on other wires made of the same material, considering their length and diameter. This makes predictions about resistance versatile and practical.

Direct Variation

Direct variation is the concept that a quantity increases or decreases in direct proportion to another quantity. For resistance, this means that if you know the resistance of a wire will increase as the length increases, you are seeing direct variation in action. This relationship holds the form \( R \propto L \), meaning:

• If \( L \) doubles, \( R \) doubles.
• Conversely, if \( L \) is halved, \( R \) is halved.

In our equation \( R = k \frac{L}{d^2} \), the \( L \) component shows this direct variation with resistance. Understanding this makes predicting how changes in wire length will affect the resistance more intuitive.

Inverse Variation

Inverse variation describes how one quantity decreases as another quantity increases. In our resistance equation, resistance varies inversely with the square of the wire's diameter. Mathematically, this is demonstrated through \( R \propto \frac{1}{d^2} \). What this means is:

• When \( d \) increases, the resistance \( R \) decreases.
• Specifically, if the diameter \( d \) is doubled, the resistance \( R \) becomes a quarter of its original value, due to the squared relationship.

In the formula \( R = k \frac{L}{d^2} \), \( d^2 \) represents this inverse variation. This relationship helps make sense of why larger diameters result in lower resistance, by distributing electrical flow more efficiently.