Problem 52 - Algebra 2

Question

For Exercises 51 and \(52,\) use the following information. In an electrical circuit, if two resistors with resistance \(R_{1}\) and \(R_{2}\) are connected in parallel as shown, the relationship between these resistances and the resulting combination resistance \(R\) is \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). A circuit with two resistors connected in parallel has an effective resistance of 25 ohms. One of the resistors has a resistance of 30 ohms. Find the resistance of the other resistor.

Step-by-Step Solution

Answer

The other resistor has a resistance of approximately 149.25 ohms.

Step 1: Understand the Formula

The formula given in the problem is for two resistors connected in parallel. It describes the relationship between the individual resistances, \(R_1\) and \(R_2\), and the equivalent resistance, \(R\), of the circuit: \[\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\]. This formula is key in solving for the unknown resistance.

Step 2: Identify Known Values

We know the effective resistance \(R\) of the circuit is 25 ohms, and one resistor’s resistance \(R_1\) is 30 ohms. We need to find the value of the second resistor \(R_2\).

Step 3: Substitute Known Values into the Formula

Substitute the known values into the formula: \[\frac{1}{25} = \frac{1}{30} + \frac{1}{R_2}\].

Step 4: Solve for the Unknown Resistance

First, find \(\frac{1}{30}\): \(\frac{1}{30} = 0.0333\). Substitute back: \[\frac{1}{25} = 0.0333 + \frac{1}{R_2}\]. Calculate \(\frac{1}{25}\): \(\frac{1}{25} = 0.04\). Set up the equation: \[0.04 = 0.0333 + \frac{1}{R_2}\], which simplifies to \[\frac{1}{R_2} = 0.04 - 0.0333\]. Calculate \(\frac{1}{R_2}\): \(\frac{1}{R_2} = 0.0067\). Invert this to get \(R_2\): \[R_2 = \frac{1}{0.0067} \approx 149.25\] ohms.

Key Concepts

Resistance Formula

When working with electrical circuits, especially those involving resistors in parallel, understanding the resistance formula is crucial. The formula at the heart of parallel resistors is:\[\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\]This equation allows for the calculation of the total or equivalent resistance, denoted as \(R\), when two resistors \(R_1\) and \(R_2\) are connected side by side in a parallel configuration. Using the formula involves the following steps:

• Convert the resistance of each resistor into its reciprocal (or the inverse), which means you are doing \(\frac{1}{\text{resistor}}\).
• Add these reciprocals together.
• Convert the sum back into the total equivalent resistance by taking the reciprocal of the result.

This process might seem challenging initially, but it becomes intuitive with practice. Remember, the total resistance in a parallel circuit is always less than the smallest individual resistance.

Circuit Analysis

Circuit analysis involves understanding how the various components in an electric circuit work together. Specifically, for resistors in parallel, it's important to analyze the roles and impacts of each component on the total circuit. Here's how you can analyze a parallel circuit:

• Identify each resistor and note its resistance value.
• Use the resistance formula to determine how each individual resistor affects the total resistance.
• Consider how changes to one resistor impact the whole circuit, particularly its voltage and current distribution.

In our initial problem, we evaluated a circuit where the total resistance was known (25 ohms), and one individual resistance was given (30 ohms). By substituting these values into our resistance formula, we systematically solved for the unknown resistor. Such analysis helps in designing electrical circuits to meet specific criteria.

Electric Circuits

Electric circuits are pathways through which electric current flows. They are composed of various elements such as resistors, capacitors, and inductors, each with a specific function. In simple terms, a resistor is used to control the flow of electric current in the circuit. In an electric circuit involving resistors in parallel:

• Each resistor has the same voltage across it but can carry different currents.
• Parallel configurations are beneficial for maintaining voltage consistency across components.
• Such circuits are commonly found in household wiring systems, where multiple appliances share the same voltage level.

Understanding electric circuits, particularly the behavior of parallel resistors, is fundamental in both industrial applications and everyday electrical systems. Properly designed circuits ensure efficiency and safety in delivering power to various loads.