Problem 51 - 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 \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}\). If \(R_{1}\) is \(x\) ohms and \(R_{2}\) is 4 ohms less than twice \(x\) ohms, write an expression for \(\frac{1}{R}\).

Step-by-Step Solution

Answer

\(\frac{1}{R} = \frac{1}{x} + \frac{1}{2x - 4}\)

Step 1: Identify Given Variables

We have two resistors: \(R_1\) is given as \(x\) ohms and \(R_2\) is given as 4 ohms less than twice \(x\) ohms. Hence, \(R_2 = 2x - 4\).

Step 2: Substitute Resistances in Formula

We know that the formula for resistors in parallel is \(\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\). Substitute \(R_1 = x\) and \(R_2 = 2x - 4\) into this formula.

Step 3: Write the Expression for \(\frac{1}{R}\)

Substituting in the values, we get \(\frac{1}{R} = \frac{1}{x} + \frac{1}{2x - 4}\).

Step 4: Simplification of the Expression

There is no further simplification necessary for this step in terms of expressing \(\frac{1}{R}\), this is the required expression with given values.

Key Concepts

Resistance Formula

In an electrical circuit, when resistors are connected in parallel, it changes the way the current flows through the circuit compared to when they are connected in series. In parallel circuits, the voltage across each resistor is the same, but the total current is the sum of the currents through each resistor. To find the equivalent resistance (a single resistor that could replace the multiple resistors without changing the current or voltage in the circuit), we use the resistance formula. The formula for two resistors is:

• \(\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2}\)

Here, \(R\) is the equivalent resistance, \(R_1\) and \(R_2\) are the resistances of the individual resistors. This formula is used because the inverse of the total resistance is equal to the sum of the inverses of each individual resistance. This results in the total resistance of a parallel circuit being less than the smallest individual resistor in that circuit. Understanding and applying this formula is essential in analyzing parallel electrical circuits.

Electrical Circuits

An electrical circuit is a pathway that allows electricity to flow. It consists of various components such as resistors, wires, batteries, and switches. These components create a closed loop, allowing electricity to flow through. Circuits can be open or closed, depending on whether the path is complete. There are two primary types of circuit arrangements: parallel and series. In a parallel circuit, all components are connected across the same two points forming a distinct loop. Each component has its own direct path to the power source. This design provides distinct advantages such as:

• The ability to maintain voltage across all components.
• Ensuring that if one resistor fails, the others can still function.
• In homes and buildings, parallel circuits allow different outlets and lights to operate independently.

Recognizing different circuit designs and their benefits is a vital part of understanding electrical electronics.

Algebraic Expressions

Algebraic expressions are a way of representing relationships using symbols and numbers. They are used extensively in mathematics to simplify and solve problems. In the context of electrical circuits, algebraic expressions allow us to symbolize relationships, like those among resistors.In this exercise, you are given the expression:

• \(R_1 = x\)
• \(R_2 = 2x - 4\)

These expressions represent resistances based on the given conditions. Algebra helps us combine them into one expression for \(\frac{1}{R}\) by substituting identified resistances into the formula:

• \(\frac{1}{R} = \frac{1}{x} + \frac{1}{2x - 4}\)

This expression now illustrates the relationship between the circuit's total resistance and individual resistances. Simplifying expressions like these often aids in identifying the behavior of electrical components, crucial for circuit design and analysis.