Identify the series combination.

Here are the solutions for the circuit problem: 1) Current through 3 \, : Step 1: Identify the series combination. The 3 \, and 5 \, resistors are connected in series. Their equivalent resistance is: R_series = 3 \, + 5 \, = 8 \, Step 2: Determine the voltage across the series combination. The 8 \, series combination is in parallel with the 2 \, resistor, and this entire parallel arrangement is connected directly across the 6.0 \, V battery. Therefore, the voltage across the 8 \, series combination is 6.0 \, V. Step 3: Calculate the current through the 3 \, resistor. Since the 3 \, resistor is part of the 8 \, series combination, the current flowing through it is the total current through that branch. Using Ohm's Law (I = (V)/(R)): I_3 = (6.0 \, V)/(8 \, ) = 0.75 \, A The current through the 3 \, resistor is 0.75 A. 2) If each cell has an internal resistance of 0.75 \, , determine the total resistance in the circuit: Step 1: Calculate the total external resistance (R_ext). The 3 \, and 5 \, resistors are in series: R_s1 = 3 \, + 5 \, = 8 \, This R_s1 is in parallel with the 2 \, resistor. The equivalent resistance of this parallel combination is: R_ext = R_s1 × 2 \, R_s1 + 2 \, = (8 \, × 2 \, )/(8 \, + 2 \, ) = (16 \, ^2)/(10 \, ) = 1.6 \, Step 2: Calculate the total internal resistance (r_total). The battery has an EMF of 6.0 \, V. Assuming standard 1.5 \, V cells, the number of cells in the battery is: n = (6.0 \, V)/(1.5 \, V) = 4 cells Each cell has an internal resistance of 0.75 \, . So, the total internal resistance for 4 cells in series is: r_total = n × r_cell = 4 × 0.75 \, = 3.0 \, Step 3: Determine the total resistance in the circuit. The total resistance in the circuit is the sum of the total external resistance and the total internal resistance: R_total = R_ext + r_total = 1.6 \, + 3.0 \, = 4.6 \, The total resistance in the circuit is $4.