Final answer:
To discover the unknown resistor value r, we establish a relationship between the power consumed when resistors are in series and parallel. A known resistor of 18.0 Ω is used, and the power ratio of 7.80 leads to a quadratic equation that provides two potential values for r.
Step-by-step explanation:
Finding the Unknown Resistor Value
To find the value of an unknown resistor r when connected in series and parallel with an 18.0 Ω resistor to an ideal battery, we must use the fact that the power consumed is 7.80 times greater when the resistors are in parallel compared to when they are in series. We'll denote the known resistance as R and set its value as 18.0 Ω.
Calculating Series and Parallel Resistances
In series, the total resistance is Rseries = R + r. In parallel, the total resistance can be found using the reciprocal formula:
Rparallel = (1/R + 1/r)−1
Using Power Relationships
Given that the power consumed Pparallel in parallel is 7.80 times that of the power consumed Pseries in series, we can set up the following relationship:
Pparallel / Pseries = (V2 / Rparallel) / (V2 / Rseries) = (Rseries / Rparallel) = 7.80
To solve for r, we manipulate the equation:
R + r = 7.80 (1/R + 1/r)−1
After finding a common denominator, factoring, and solving the quadratic equation, there will be two possible results for r, which are the two values of the unknown resistor.