Problem 116 - College Algebra with Modeling and Visualization

Question

The electrical resistance of a wire varies directly with its length. If a 255 -foot wire has a resistance of 1.2 ohms, find the resistance of 135 feet of the same type of wire. Interpret the constant of proportionality in this situation.

Step-by-Step Solution

Answer

The resistance of a 135-foot wire is approximately 0.6345 ohms, with a constant of proportionality of 0.0047 ohms per foot.

Step 1: Understanding Direct Variation

The problem states that the resistance of the wire varies directly with its length. This means if the length of the wire increases, the resistance increases proportionally. Mathematically, this is written as \( R = kL \), where \( R \) is the resistance, \( L \) is the length, and \( k \) is the constant of proportionality.

Step 2: Finding the Constant of Proportionality

We are given that a 255-foot wire has a resistance of 1.2 ohms. We can use the equation \( R = kL \) to find \( k \). Plug in the values: \( 1.2 = k \times 255 \). Solve for \( k \) by dividing both sides by 255: \( k = \frac{1.2}{255} \approx 0.0047 \).

Step 3: Calculating Resistance for a 135-Foot Wire

Now that we have \( k = 0.0047 \), we can calculate the resistance for a 135-foot wire using the same relationship: \( R = kL \). Substitute \( L = 135 \) and \( k = 0.0047 \) into the equation: \( R = 0.0047 \times 135 \). This gives \( R \approx 0.6345 \) ohms.

Key Concepts

Proportionality

Proportionality is a fundamental concept in mathematics that describes a consistent and direct relationship between two quantities. If one quantity increases, and the other does as well, they are said to be directly proportional. This means you can express one quantity as a multiple of the other, involving a constant of proportionality. In this scenario, you can always rely on the pattern: if the length of the wire increases, its resistance increases too.
To visualize this, think of an equation like \( R = kL \), where \( R \) is the resistance, \( L \) is the length, and \( k \) represents our constant of proportionality. This constant informs how drastically resistance changes when length does:

• For a longer wire, \( R \) would be higher with a fixed \( k \).
• However, if you cut the length in half, the resistance also halves.

Knowing the proportional relationship helps to predict unknown values if you know the constant value \( k \).

Algebraic Equations

Algebraic equations allow us to find unknown values using mathematical expressions. In direct variation problems like this, we start with an equation, \( R = kL \), that relates resistance to length, providing clues on how to compute unmeasured aspects.
Let's break it down:

• First, we compute the constant \( k \) using known values — in this case, a 255-foot wire offering 1.2 ohms resistance: \( 1.2 = k \times 255 \).
• To find \( k \), just divide both sides by 255: \( k = \frac{1.2}{255} \).
• This yields \( k \approx 0.0047 \), telling us how much resistance increases per foot.

After deriving \( k \), we substitute values into the original equation to find unknown resistances. Simply input different wire lengths, like 135 feet, and solve \( R = 0.0047 \times 135 \) to find the resistance.

Problem Solving

Approaching problem-solving systematically can transform complex questions into simpler ones. Here’s a structured way to tackle problems involving direct variation and algebraic equations.
First, identify what is being asked and the conditions provided. In this exercise, the problem needs us to determine how wire length affects resistance. Here's an effective approach:

• Comprehend the concept of direct variation and how it connects resistance and length via a constant.
• Calculate this proportionality constant, employing provided data. With this constant, future unknowns become predictable.
• Apply your constant for verbose scenarios, accurately computing missing data through straightforward algebra.

This approach isn’t limited to wires and resistance alone, but extends to many situations involving consistent relationships with more than one variable.