Problem 55 - Elementary Algebra

Question

In an experiment, an electronics student finds that the total resistance of a series circuit with two resistors is 78 ohms. The total resistance of a parallel circuit with the same resistors is \(8.7\) ohms. The resistance of each resistor in ohms is a solution of the equation \(x^{2}-78 x+678.6=0\). Find the resistance of each resistor. Round to the nearest whole number.

Step-by-Step Solution

Answer

The resistances are approximately 68 ohms and 10 ohms.

Step 1: Identify the equation

The given equation is a quadratic equation: \[x^{2} - 78x + 678.6 = 0\]We'll need to solve this equation to find the resistances.

Step 2: Use the quadratic formula

The quadratic formula is \[x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\]For the given equation, the coefficients are: - \(a = 1\)- \(b = -78\)- \(c = 678.6\)

Step 3: Calculate the discriminant

The discriminant (\(D\)) is calculated as: \[D = b^2 - 4ac\]Inserting the values: \[D = (-78)^2 - 4(1)(678.6)\]\[D = 6084 - 2714.4\]\[D = 3369.6\]

Step 4: Solve for x using the quadratic formula

Substitute \(D\) back into the quadratic formula: \[x = \frac{78 \pm \sqrt{3369.6}}{2}\]Calculate the square root and solve: \[x = \frac{78 \pm 58.05}{2}\]Thus, we get two solutions: \[x_1 = \frac{78 + 58.05}{2} = 68.025\]\[x_2 = \frac{78 - 58.05}{2} = 9.975\]

Step 5: Round the solutions

Round each solution to the nearest whole number: \[x_1 \approx 68\] \[x_2 \approx 10\]Therefore, the resistances are 68 ohms and 10 ohms.

Key Concepts

Resistance in Series Circuits

In a series circuit, resistors are connected end-to-end. The total resistance in a series circuit is simply the sum of the individual resistances. This occurs because the current has only one path to follow, and it encounters the full resistance of each resistor sequentially.
This principle makes it straightforward to determine the total resistance:

• If you have two resistors, R1 and R2, the total resistance (R_total) is: \[ R_{total} = R_1 + R_2 \]

The series circuit in our exercise has a total resistance of 78 ohms. Knowing this allows us to set up the quadratic equation to find the individual resistances.

Resistance in Parallel Circuits

A parallel circuit has resistors connected across the same two points, creating multiple paths for the current to flow through. The total resistance in a parallel circuit is less than the smallest individual resistance because the current is divided among the multiple paths.
The formula for calculating the total resistance (R_total) of two resistors, R1 and R2, in parallel is:

• \[ \frac{1}{R_{total}} = \frac{1}{R_1} + \frac{1}{R_2} \]

In the given exercise, the total resistance for the parallel circuit is 8.7 ohms. This setting helps us to anticipate the interaction between the resistors and to solve for their individual resistances when applied to the quadratic formula.

Quadratic Formula

The quadratic formula is a powerful tool for solving equations of the form:

• \[ ax^2 + bx + c = 0 \]

In this case, the coefficients are a = 1, b = -78, and c = 678.6. The formula to find the roots (solutions) of the quadratic equation is:

• \[ x = \frac{-b \, \pm\, \sqrt{ b^2 - 4ac }}{2a} \]

Following the steps in the solution:

• Calculate the discriminant \[ D = b^2 - 4ac \], producing \[ D = 3369.6 \]
• Substituting D back into the quadratic formula: \[ x = \frac{78 \, \pm\, \sqrt{3369.6 }}{2} \]
• Solving, we get roots: \[ x_1 \approx 68 \quad\text{and}\quad x_2 \approx 10 \]

Therefore, the individual resistances of the resistors are approximately 68 ohms and 10 ohms. The quadratic formula helps simplify solving polynomial equations, converting a complex problem into manageable steps.