To determine the resistances of the two resistors, we can use the formulas for resistors in series and in parallel.
In a series circuit, the total resistance (R_total) is the sum of the individual resistances (R1 and R2):
R_total = R1 + R2
In a parallel circuit, the reciprocal of the total resistance (1/R_total) is the sum of the reciprocals of the individual resistances:
1/R_total = 1/R1 + 1/R2
Given that the current from the battery (I) is 1.99 A in the series circuit and 10.7 A in the parallel circuit, we can use Ohm's Law to relate the current, voltage (V), and resistance:
I = V/R
Let's solve for the resistances step by step:
Step 1: Determine the resistance in the series circuit.
Using Ohm's Law, we have:
1.99 A = 12.0 V / R_total
R_total = 12.0 V / 1.99 A
R_total ≈ 6.03 Ω
Step 2: Determine the resistance in the parallel circuit.
Using Ohm's Law, we have:
10.7 A = 12.0 V / R_total
R_total = 12.0 V / 10.7 A
R_total ≈ 1.12 Ω
Step 3: Solve for the individual resistances.
For the series circuit:
R_total = R(smaller) + R(larger)
6.03 Ω = R(smaller) + R(larger)
For the parallel circuit:
1/R_total = 1/R(smaller) + 1/R(larger)
1/1.12 Ω = 1/R(smaller) + 1/R(larger)
We have two equations and two unknowns (R(smaller) and R(larger)). We can solve this system of equations using algebraic methods or substitution.
One possible solution is:
R(smaller) = 4 Ω
R(larger) = 2.03 Ω
Therefore, the two resistances are R(smaller) = 4 Ω and R(larger) = 2.03 Ω.