To solve the problem of finding the image distance behind a concave mirror, we will use the mirror formula and the information provided in the question. Let's break it down step-by-step. ### Step 1: Understand the Mirror Formula The mirror formula relates the object distance (u), the image distance (v), and the focal length (f) of a mirror. The formula is given by: \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] ### Step 2: Determine the Focal Length The radius of curvature (R) of the mirror is given as 1 meter. The focal length (f) of a mirror is related to its radius of curvature by the formula: \[ f = \frac{R}{2} \] For our case: \[ f = \frac{1 \, \text{m}}{2} = 0.5 \, \text{m} \] ### Step 3: Identify the Object Distance The object distance (u) is the distance from the object (the vehicle behind the car) to the mirror. Since the object is in front of the mirror, we take this distance as negative in the mirror formula convention: \[ u = -4 \, \text{m} \] ### Step 4: Substitute Values into the Mirror Formula Now we can substitute the values of f and u into the mirror formula: \[ \frac{1}{0.5} = \frac{1}{v} + \frac{1}{-4} \] This simplifies to: \[ 2 = \frac{1}{v} - \frac{1}{4} \] ### Step 5: Solve for Image Distance (v) To isolate \(\frac{1}{v}\), we can rearrange the equation: \[ \frac{1}{v} = 2 + \frac{1}{4} \] To add these fractions, we need a common denominator. The common denominator for 2 (which is \(\frac{8}{4}\)) and \(\frac{1}{4}\) is 4: \[ \frac{1}{v} = \frac{8}{4} + \frac{1}{4} = \frac{9}{4} \] Now, taking the reciprocal to find v: \[ v = \frac{4}{9} \, \text{m} \] ### Step 6: Interpret the Result The image distance (v) is positive, which indicates that the image is formed on the same side as the object, which is typical for a virtual image produced by a concave mirror when the object is placed within the focal length. ### Conclusion: Correct Option The calculated image distance is \( \frac{4}{9} \, \text{m} \), which corresponds to option B. ### Why Other Options Are Incorrect - **Option A (8/7)**: This value does not correspond to any calculation based on the mirror formula and is not derived from the given parameters. - **Option C (9/2)**: This value is too large and does not fit within the context of the problem, as it suggests an image distance that is not possible given the object distance and focal length. - **Option D (4/7)**: This value is also incorrect as it does not match our calculations and does not satisfy the mirror formula. ### Revision Summary - The mirror formula is \(\frac{1}{f} = \frac{1}{v} + \frac{1}{u}\). - The focal length of a mirror is half the radius of curvature: \(f = \frac{R}{2}\). - Object distance is taken as negative in the mirror formula convention. - The image distance for the given problem is \(v = \frac{4}{9} \, \text{m}\), corresponding to option B.