To determine the two resistances, let's start by using Ohm's law to calculate the resistance values in the two given scenarios.
1. Series Connection:
When two resistors are connected in series, the total resistance (R_total) is the sum of the individual resistors' values.
Based on Ohm's law (V = I * R), where V is the voltage, I is the current, and R is the resistance, we can derive the equation for total resistance in a series circuit as follows:
R_total = R(smaller) + R(larger)
We know that the current (I) is 1.99 A and the voltage (V) is 12.0 V.
Rearranging the equation to solve for the total resistance, we have:
R_total = V / I
R_total = 12.0 V / 1.99 A
Calculating this, the total resistance in the series circuit is approximately 6.03 Ω.
2. Parallel Connection:
When two resistors are connected in parallel, the inverse of the total resistance (1/R_total) is equal to the sum of the inverses of the individual resistances.
Using the same equation (V = I * R), we get:
R(smaller) = V / I
R(larger) = V / I
Given that the current (I) is 10.7 A and the voltage (V) is 12.0 V, we can solve for the individual resistances:
1/R_total = 1/R(smaller) + 1/R(larger)
1/R_total = 1 / (V / I) + 1 / (V / I)
1/R_total = 1 / (12.0 V / 10.7 A) + 1 / (12.0 V / 10.7 A)
After evaluating this expression, we find that the inverse of the total resistance (1/R_total) is approximately 1.68 S (Siemens).
Finally, to find the individual resistance values, we can take the reciprocals of the inverse total resistance:
R(smaller) = 1 / (1.68 S)
R(larger) = 1 / (1.68 S)
Evaluating these equations, we get the individual resistances approximately 0.595 Ω for the smaller resistor and 0.595 Ω for the larger resistor.
Therefore, the two resistance values are 0.595 Ω (smaller) and 0.595 Ω (larger) when rounded to three decimal places.