Problem 77 - Calculus Early Transcendentals: Pearson New International Edition

Question

The formula \(\frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}}\) gives the total resistance \(R\) in an electric circuit due to three resistances, \(R_{1}, R_{2}\), and \(R_{3}\), connected in parallel. If \(10 \leq R_{1} \leq 20,20 \leq R_{2} \leq 30\), and \(30 \leq R_{3} \leq 40\), find the range of values for \(R\)

Step-by-Step Solution

Answer

The range of total resistance \( R \) is approximately 5.45 to 9.23 ohms.

Step 1: Understanding the Formula

The formula \( \frac{1}{R}=\frac{1}{R_{1}}+\frac{1}{R_{2}}+\frac{1}{R_{3}} \) represents the total resistance \( R \) for resistors in parallel, meaning the reciprocals of individual resistances add up to the reciprocal of the total resistance.

Step 2: Determine Bounds for Reciprocal Resistances

To find the range of total resistance, compute the sum of reciprocals of minimum and maximum values of \( R_1, R_2, \) and \( R_3 \). This will help determine the extremities for the reciprocal of \( R \).

Step 3: Calculate Reciprocal Lower Bound

For the lower bound of \( \frac{1}{R} \), find the sum using maximum values of \( R_1, R_2, \) and \( R_3 \):\[ \frac{1}{R} = \frac{1}{10} + \frac{1}{20} + \frac{1}{30} \]\[ \frac{1}{R} = 0.1 + 0.05 + 0.0333 \approx 0.1833 \]

Step 4: Calculate Reciprocal Upper Bound

For the upper bound of \( \frac{1}{R} \), find the sum using minimum values of \( R_1, R_2, \) and \( R_3 \):\[ \frac{1}{R} = \frac{1}{20} + \frac{1}{30} + \frac{1}{40} \]\[ \frac{1}{R} = 0.05 + 0.0333 + 0.025 \approx 0.1083 \]

Step 5: Find Range of Total Resistance

The range of total resistance \( R \) is determined by calculating the reciprocals of the bounds for \( \frac{1}{R} \):\[ R_{\text{min}} = \frac{1}{0.1833} \approx 5.45 \]\[ R_{\text{max}} = \frac{1}{0.1083} \approx 9.23 \]Thus, the resistance \( R \) can range from about 5.45 to 9.23 ohms.

Key Concepts

electric circuit analysis

Electric circuit analysis is essential to understanding how different electrical components connect to form a working circuit. In this context, analyzing an electric circuit involves understanding how the total resistance is impacted when resistors are connected in parallel. Unlike series connections where resistances add up, parallel connections require us to think a little differently due to the addition of reciprocals of resistances.

When resistors are in parallel, the voltage across each resistor is the same. However, the current can differ. This impacts how we calculate total resistance, significantly lowering it compared to individual resistors. The formula \(\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}\) helps in determining this reduced equivalent resistance, essential for efficient circuit design.
Understanding these concepts in electric circuit analysis is crucial, especially in designing circuits with specific resistance requirements like those in calculating range through parallel resistor combinations.

reciprocal calculations

Reciprocal calculations are a vital part of working with parallel circuits to find the equivalent resistance. Knowing how to correctly calculate the reciprocals of resistance ensures that you get an accurate total resistance value.

When dealing with parallel resistors, you calculate the reciprocals of each individual resistance and then sum them up. This total is the reciprocal of the total resistance \(R\):
\[\frac{1}{R} = \frac{1}{R_{1}} + \frac{1}{R_{2}} + \frac{1}{R_{3}}\]
Working with reciprocals might seem complex at first, but it’s like handling fractions.

• Add up all the reciprocals of individual resistances to find the reciprocal of the total resistance.
• Finally, find \( R \) by taking the reciprocal of this added value, essentially flipping it back to get the resistance.

This method simplifies what’s happening in a circuit, allowing you to effectively analyze and work with complex circuits.

resistor combinations

Resistor combinations in parallel circuits offer unique advantages in circuit design. Such combinations provide a convenient way to tailor the overall resistance required by a specific part of a circuit. As discussed, when resistors are connected in parallel, the total resistance is always less than any of the individual resistors used in the combination.

This combination strategy serves several purposes:

• It can help achieve precise control over the circuit’s behavior by adjusting the total resistance to a desired level.
• It can ensure greater stability and reliability as a decrease in one resistance value won't significantly affect the overall resistance.

By effectively using resistor combinations, circuit designers can ensure optimal performance in both simple and complex circuits. It's a fundamental skill that strengthens one's ability to work with electrical and electronic systems efficiently.