To solve the problem of finding the object distance for a concave mirror that forms a real image twice as large as the object, we will use the mirror formula and magnification concepts. Let's break this down step-by-step. ### Step 1: Understand the Given Information - **Radius of Curvature (R)**: 40 cm - **Focal Length (f)**: The focal length of a concave mirror is given by the formula: \[ f = \frac{R}{2} \] Therefore, the focal length is: \[ f = \frac{40 \, \text{cm}}{2} = 20 \, \text{cm} \] - **Magnification (m)**: The problem states that the image is twice as large as the object, which means: \[ m = -2 \] (The negative sign indicates that the image is real and inverted.) ### Step 2: Use the Magnification Formula The magnification for mirrors is given by the formula: \[ m = -\frac{v}{u} \] where: - \( v \) = image distance - \( u \) = object distance Since we know \( m = -2 \), we can rearrange the formula to find the relationship between \( v \) and \( u \): \[ -2 = -\frac{v}{u} \implies v = 2u \] ### Step 3: Use the Mirror Formula The mirror formula relates the object distance \( u \), the image distance \( v \), and the focal length \( f \): \[ \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \] Substituting \( f = 20 \, \text{cm} \) and \( v = 2u \) into the mirror formula gives: \[ \frac{1}{20} = \frac{1}{2u} + \frac{1}{u} \] ### Step 4: Simplify the Equation To combine the terms on the right side, we need a common denominator: \[ \frac{1}{2u} + \frac{1}{u} = \frac{1 + 2}{2u} = \frac{3}{2u} \] Now, substituting this back into the mirror formula: \[ \frac{1}{20} = \frac{3}{2u} \] ### Step 5: Solve for \( u \) Cross-multiplying gives: \[ 2u = 20 \times 3 \implies 2u = 60 \implies u = 30 \, \text{cm} \] ### Conclusion The object distance \( u \) is **30 cm**. Therefore, the correct option is **B**. ### Step 6: Explanation of Other Options - **Option A (10 cm)**: If the object distance were 10 cm, the image distance would be calculated using the mirror formula, leading to a magnification that does not equal -2. Thus, this option is incorrect. - **Option C (40 cm)**: If the object distance were 40 cm, the image distance would also not yield a magnification of -2. This option does not satisfy the conditions of the problem. - **Option D (60 cm)**: If the object distance were 60 cm, the resulting image distance would again not yield a magnification of -2. This option is also incorrect. ### Revision Summary - The focal length of a concave mirror is half the radius of curvature. - Magnification relates the image distance and object distance: \( m = -\frac{v}{u} \). - The mirror formula connects object distance, image distance, and focal length: \( \frac{1}{f} = \frac{1}{v} + \frac{1}{u} \). - For a real image that is twice the size of the object, the object distance is found to be 30 cm.