Problem 9 - Algebra 2

Question

ELECTRICITY For Exercises \(9-12\) , use the following information. The current \(I\) in amperes in an electrical circuit with three resistors in series is given by the equation \(I=\frac{V}{R_{1}+R_{2}+R_{3}},\) where \(V\) is the voltage in volts in the circuit and \(R_{1}, R_{2},\) and \(R_{3}\) are the resistances in ohms of the three resistors. Let \(R_{1}\) be the independent variable, and let I be the dependent variable. Graph the function if \(V=120\) volts, \(R_{2}=25\) ohms, and \(R_{3}=75\) ohms.

Step-by-Step Solution

Answer

Graph the function \(I = \frac{120}{R_1 + 100}\) with \(R_1\) on x-axis and \(I\) on y-axis.

Step 1: Identify the Formula

We are given the formula for current in a series circuit with three resistors: \(I = \frac{V}{R_1 + R_2 + R_3}\).

Step 2: Substitute Known Values

Substitute the known values for voltage \(V = 120\, \text{V}\), \(R_2 = 25\, \text{ohms}\), and \(R_3 = 75\, \text{ohms}\) into the equation: \[I = \frac{120}{R_1 + 25 + 75}\].

Step 3: Simplify Expression

Combine the constant resistances \(R_2\) and \(R_3\) in the denominator: \(R_2 + R_3 = 100\). The expression becomes \[I = \frac{120}{R_1 + 100}\].

Step 4: Determine Behavior of the Function

Determine how \(I\) changes as \(R_1\) changes. As \(R_1\) increases, the denominator increases, so \(I\) decreases. Conversely, if \(R_1\) decreases, \(I\) increases.

Step 5: Graph the Function

With \(R_1\) on the x-axis and \(I\) on the y-axis, graph the function \(I = \frac{120}{R_1 + 100}\). This is a hyperbola that asymptotically approaches the x-axis as \(R_1\) increases and rises steeply as \(R_1\) approaches zero from the right.

Key Concepts

Series Circuits

A series circuit is a type of electrical circuit where components are connected end-to-end, creating a single path for electrical current to flow. In other words, electrons travel in a continuous loop, passing through each circuit element one after the other. This is different from a parallel circuit where multiple paths exist for the current to flow. In a series circuit, the same current flows through all components because there is only one pathway for the charge to move.

• All the resistors in a series circuit share the same current.
• However, the total voltage across the circuit is the sum of the voltage drops across each resistor.
• Ohm's Law, which states that the voltage is equal to the current times the resistance (\( V = I imes R \)), applies to the entire circuit.

In this exercise, the focus is on understanding how total resistance affects the current in a series circuit with three resistors in sequence. The formula given helps determine how the individual resistances together change the total resistance and current flowing through the circuit.

Graphing Equations

Graphing equations is a vital mathematical skill, particularly when analyzing functions like the current in a series circuit. To graph an equation, you need to understand its form, dependent and independent variables, and the behavior on a graph.
In this exercise, the graph represents the relationship between the resistance of one of the resistors (\( R_1 \)) and the current (\( I \)) flowing through the circuit.

• The x-axis represents the independent variable (\( R_1 \)).
• The y-axis represents the dependent variable (\( I \)).
• Since the formula is \( I = \frac{120}{R_1 + 100} \), the graph of this function will show a hyperbolic shape.

Hyperbolas are curved lines that approach the axes without touching them. In this context, as \( R_1 \) increases, the curve approaches zero, illustrating that the current decreases.

Dependent and Independent Variables

When dealing with equations and graphs, it’s important to distinguish between dependent and independent variables. This distinction helps us understand how one variable affects another.

• The independent variable is the one we change or control to see how it affects another variable. In this scenario, it is \( R_1 \), a resistor's resistance.
• The dependent variable is the one that responds to the independent variable. Here, that is the current \( I \), which changes based on the resistance \( R_1 \) and others.

In science, understanding these terms lets us create more accurate models of natural phenomena. By knowing what influences what, we can predict and analyze outcomes effectively. In our circuit example, \( R_1 \) affects the total resistance and, consequently, alters the current \( I \). The ability to graph these changes shows the correlation between resistance in a series circuit and the resulting current.