A lens forms an image at 30cm when the object is placed at 40cm. The image cannot be caught on the screen. Calculate image height when the object height is 4cm.

To solve this problem, we need to use the lens formula and the magnification formula.

1. The lens formula is 1/f = 1/v - 1/u. Here, v is the image distance, u is the object distance, and f is the focal length of the lens. We are given that v = -30 cm (the negative sign indicates that the image is virtual and on the same side as the object) and u = -40 cm (the negative sign indicates that the object is on the same side as the light source). We can substitute these values into the lens formula to find f.

2. Substituting the given values into the lens formula, we get 1/f = 1/(-30) - 1/(-40) = -1/120. Therefore, f = -120 cm.

3. The magnification formula is m = -v/u = h'/h, where h' is the height of the image and h is the height of the object. We are given that h = 4 cm. We can substitute the given values into the magnification formula to find h'.

4. Substituting the given values into the magnification formula, we get m = -(-30)/(-40) = -3/4. Therefore, h' = m * h = -3/4 * 4 = -3 cm.

5. The negative sign indicates that the image is inverted. Therefore, the height of the image is 3 cm and it is inverted.