To determine the power dissipated in a resistor when an alternating current (AC) flows through it, we can use the formula for power in an AC circuit. Let's break down the problem step-by-step. ### Given Information: - The current is given by the equation: \[ I(t) = 10 \sin(120\pi t) \] - The resistance \( R \) is \( 12 \, \Omega \). ### Step 1: Identify the RMS Current In AC circuits, we often use the root mean square (RMS) value of the current to calculate power. The RMS value of a sinusoidal current is given by: \[ I_{\text{RMS}} = \frac{I_0}{\sqrt{2}} \] where \( I_0 \) is the peak current. From the equation \( I(t) = 10 \sin(120\pi t) \), we can see that the peak current \( I_0 \) is \( 10 \, A \). Now, we can calculate the RMS current: \[ I_{\text{RMS}} = \frac{10}{\sqrt{2}} \approx 7.07 \, A \] ### Step 2: Calculate the Power Dissipated The power dissipated in a resistor in an AC circuit can be calculated using the formula: \[ P = I_{\text{RMS}}^2 \cdot R \] Substituting the values we have: \[ P = (7.07)^2 \cdot 12 \] Calculating \( (7.07)^2 \): \[ (7.07)^2 \approx 50 \] Now, substituting this back into the power formula: \[ P = 50 \cdot 12 = 600 \, W \] ### Conclusion Thus, the power dissipated in the resistor is **600 W**. Therefore, the correct option is **B**. ### Explanation of Other Options: - **Option A (1200 W)**: This value could be mistakenly calculated by using the peak current directly without converting to RMS. The formula \( P = I_0^2 \cdot R \) would yield \( P = 10^2 \cdot 12 = 1200 \, W \), which is incorrect for AC circuits. - **Option C (120 W)**: This value does not correspond to any standard calculation for power in this scenario. It may arise from a misunderstanding of the relationship between current and power. - **Option D (30 W)**: This value is also incorrect and does not relate to the calculations we performed. It could be a miscalculation or misunderstanding of the formulas involved. ### Summary for Revision: - The RMS current for a sinusoidal AC current is calculated as \( I_{\text{RMS}} = \frac{I_0}{\sqrt{2}} \). - Power in an AC circuit is calculated using \( P = I_{\text{RMS}}^2 \cdot R \). - Always ensure to use RMS values for AC calculations to avoid errors. - The correct power dissipated in a 12Ω resistor with a peak current of 10A is 600W.