Answer :
Final answer:
The resistance of a 75-W headlight is 7.68 ohms and the resistance of a 100-W headlight is 5.76 ohms. Connected in parallel, the total resistance seen by the battery is approximately 3.29 ohms.
Explanation:
To find the resistance of each headlight, we use the formula for power (P = V2/R) and solve for R, the resistance. Therefore, R = V2/P. Since the voltage (V) across each headlight is 24 volts, the resistance of a 75-W headlight is R = (24 V)2/75 W, and the resistance of the 100-W headlight is R = (24 V)2/100 W. Calculating these gives us:
Resistance of 75-W headlight: R = 242/75 = 576/75 = 7.68 ohms. Resistance of the 100-W headlight: R = 242/100 = 576/100 = 5.76 ohms. In a parallel circuit, the total resistance (Rtotal) is given by the reciprocal of the sum of the reciprocals of each resistance (1/Rtotal = 1/R1 + 1/R2). The calculation is as follows:
Calculate the reciprocal of each resistance: 1/7.68 = 0.1302 and 1/5.76 = 0.1736. Add the reciprocals: 0.1302 + 0.1736 = 0.3038. Find the reciprocal of the total to get Rtotal: 1/0.3038 ≈ 3.29 ohms. Therefore, the total resistance seen by the battery is approximately 3.29 ohms.
