Let's solve the problem **step by step**: --- ## **Step 1: Write Down the Given Data** - Electric field magnitude, \( E = 422 \, \text{N/C} \) - Angle with the surface, \( \theta = 71.^\circ \) - Area of the surface, \( A = 3.7 \, \text{m}^2 \) --- ## **Step 2: Recall the Formula for Electric Flux** The electric flux (\( \Phi_E \)) through a surface is given by: \[ \Phi_E = E \cdot A \cdot \cos\theta \] Where: - \( E \) = magnitude of electric field - \( A \) = area of the surface - \( \theta \) = angle between the electric field and the **normal** to the surface --- ## **Step 3: Check the Angle** The figure and text say the angle is **with the surface**, not the normal. The angle with the **normal** (\( \theta_n \)) is: \[ \theta_n = 90^\circ - 71.^\circ = 19.^\circ \] --- ## **Step 4: Plug in the Values** \[ \Phi_E = E \cdot A \cdot \cos(\theta_n) \] \[ \Phi_E = 422 \times 3.7 \times \cos(19.^\circ) \] --- ## **Step 5: Calculate \( \cos(19.^\circ) \)** \[ \cos(19.^\circ) \approx .9455 \] --- ## **Step 6: Calculate the Flux** \[ \Phi_E = 422 \times 3.7 \times .9455 \] \[ = 1561.4 \times .9455 \] \[ = 1476.2 \] --- ## **Step 7: Round to the Nearest Whole Number** \[ \boxed{1476} \] --- ## **Final Answer** \[ \boxed{1476 \ \text{N} \cdot \text{m}^2/\text{C}} \] --- **Explanation:** We used the formula for electric flux, converted the given angle to the angle with the normal, and calculated the result step by step.