Question

From Ohm's Law for circuits, it follows that the total resistance \(R_{\text {tot }}\) of two components hooked in parallel is given by the equation $$ R_{\mathrm{tot}}=\frac{R_{1} R_{2}}{R_{1}+R_{2}} $$ where \(R_{1}\) and \(R_{2}\) are the individual resistances. (a) Let \(R_{1}=10 \mathrm{ohms}\), and graph \(R_{\text {tot }}\) as a function of \(R_{2}\). (b) Find and interpret any asymptotes of the graph obtained in part (a). (c) If \(R_{2}=2 \sqrt{R_{1}},\) what value of \(R_{1}\) will yield an \(R_{\text {tot }}\) of 17 ohms?

Short answer

The graph of \( R_{\text{tot}} \) against \( R_2 \) has a horizontal asymptote at 10 ohms. \( R_1 = 289 \) ohms will yield \( R_{\text{tot}} = 17 \) ohms when \( R_2 = 2 \sqrt{R_1} \).

Step-by-step solution

Substitute values and equation setup

Given Ohm's Law for total resistance in parallel, the equation is \[ R_{\text{tot}} = \frac{R_1 R_2}{R_1 + R_2} \] For part (a), substitute \( R_1 = 10 \text{ ohms} \). The equation becomes \[ R_{\text{tot}} = \frac{10 R_2}{10 + R_2} \]

Graphing the function

Plot the function \( R_{\text{tot}} = \frac{10 R_2}{10 + R_2} \) using various values of \( R_2 \). For example, evaluate and plot points for \( R_2 = 1, 2, 5, 10, 20, 50 \). The graph will show how \( R_{\text{tot}} \) changes as \( R_2 \) increases.

Identify Asymptotes

Analyze the behavior of the graph from Step 2. As \( R_2 \) approaches 0, \( R_{\text{tot}} \) approaches 0. As \( R_2 \) approaches infinity, \( R_{\text{tot}} \) approaches 10 ohms. Thus, the vertical asymptote is \( R_2 = 0 \) and the horizontal asymptote is \( R_{\text{tot}} = 10 \text{ ohms} \).

Solve for specific resistance

In part (c), given \( R_{\text{tot}} = 17 \text{ ohms} \) and \( R_2 = 2 \sqrt{R_1} \), substitute these into the formula: \[ 17 = \frac{R_1 (2 \sqrt{R_1})}{R_1 + 2 \sqrt{R_1}} \] Simplify and solve for \( R_1 \). Multiply both sides by \( R_1 + 2 \sqrt{R_1} \) : \[ 17 (R_1 + 2 \sqrt{R_1}) = 2 R_1 \sqrt{R_1} \] Distribute and solve for \( R_1 \): \[ 17R_1 + 34 \sqrt{R_1} = 2 R_1 \sqrt{R_1} \] Divide both sides by \( \sqrt{R_1} \) : \[ 17 \sqrt{R_1} + 34 = 2 R_1 \] Rearrange and solve the quadratic equation: \[ 2R_1 - 17 \sqrt{R_1} - 34 = 0 \] Let \( x = \sqrt{R_1} \), solve \( 2x^2 - 17x - 34 = 0 \). Use the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \] where \( a = 2 \), \( b = -17 \), and \( c = -34 \). \[ x = \frac{17 \pm \sqrt{289 + 272}}{4} = \frac{17 \pm \sqrt{561}}{4} \] Choose the positive root and find \( R_1 = x^2 \). Therefore, the value of \( R_1 \) is the square of the positive value of \( x\).

Key concepts

Parallel Resistance

In electrical circuits, resistors can be connected in various ways, and one common method is a parallel configuration. When resistors are connected in parallel, the total resistance can be found using the formula: \[R_{\text{tot}} = \frac{R_1 R_2}{R_1 + R_2} \]This formula tells us how the total resistance, \( R_{\text{tot}} \), is affected by the individual resistances \( R_1 \) and \( R_2 \). For example, if \( R_1 = 10 \) ohms and you want to find \( R_{\text{tot}} \) for different values of \( R_2 \), you plug the values into the formula. As \( R_2 \) increases, \( R_{\text{tot}} \) will change. This behavior highlights how resistances in parallel share the load and result in a total resistance that is always less than the smallest individual resistor. Understanding this concept is crucial for designing circuits with specific resistance values.

Graphing Functions

Graphing functions allows us to visualize their behavior. For our equation \( R_{\text{tot}} = \frac{10 R_2}{10 + R_2} \), graphing this as a function of \( R_2 \) shows how \( R_{\text{tot}} \) changes as \( R_2 \) varies. When creating the graph, we can plot points for different values of \( R_2 \) such as 1, 2, 5, 10, 20, and 50. This offers a visual representation of the relationship between \( R_{\text{tot}} \) and \( R_2 \). Visualizing the function helps students understand how increasing the value of one variable impacts the other, especially in terms of resistance in parallel circuits.

Asymptotes

Asymptotes are important for understanding the behavior of functions at extreme values. For the function \( R_{\text{tot}} = \frac{10 R_2}{10 + R_2} \), we can identify two notable asymptotes:

• Vertical Asymptote: As \( R_2 \) approaches 0, \( R_{\text{tot}} \) also approaches 0. This indicates a vertical asymptote at \( R_2 = 0 \).
• Horizontal Asymptote: As \( R_2 \) becomes very large (approaches infinity), \( R_{\text{tot}} \) approaches 10 ohms, revealing a horizontal asymptote at \( R_{\text{tot}} = 10 \) ohms.

Understanding asymptotes helps predict the behavior of the resistance in parallel circuits at these extreme values, especially helpful when designing circuits for high and low resistance limits.

Quadratic Equation Solving

Solving quadratic equations is a crucial skill in math and physics. In the given problem, we set \( R_{\text{tot}} = 17 \) ohms and \( R_2 = 2 \sqrt{R_1} \) leading to the equation: \[ 17 = \frac{R_1 (2 \sqrt{R_1})}{R_1 + 2 \sqrt{R_1}} \]To solve for \( R_1 \), we rearrange and simplify:

• \( 17 (R_1 + 2 \sqrt{R_1}) = 2 R_1 \sqrt{R_1} \)
• \( 17 R_1 + 34 \sqrt{R_1} = 2 R_1 \sqrt{R_1} \)
• Divide by \( \sqrt{R_1} \): \( 17 \sqrt{R_1} + 34 = 2 R_1 \)
• Let \( x = \sqrt{R_1} \): \( 2x^2 - 17x - 34 = 0 \)

We solve this quadratic using the quadratic formula: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Substituting \(a = 2\), \(b = -17\), and \(c = -34\), we find: \[ x = \frac{17 \pm \sqrt{289 + 272}}{4} = \frac{17 \pm \sqrt{561}}{4} \]
Choosing the positive root and then squaring it gives \( R_1 \). This example shows how to approach quadratic equations logically and systematically.