Answer: -17.28 cm
To solve this problem, we can use the lens formula:
1/f = 1/do + 1/di
where f is the focal length, do is the object distance, and di is the image distance. We are given the object distance (do = 56.0 cm) and the distance between the object and the image (31.0 cm).
First, we need to determine the image distance (di). Since the lens is diverging, the image will be formed on the same side as the object. Therefore, we can calculate the image distance by subtracting the distance between the object and the image from the object distance:
di = do - 31.0 cm = 56.0 cm - 31.0 cm = 25.0 cm
Now we can use the lens formula to find the focal length:
1/f = 1/do + 1/di
1/f = 1/56.0 cm + 1/25.0 cm
To calculate 1/f, we need to add the two fractions:
1/f = (25 + 56) / (56 * 25) = 81 / 1400
Now, take the reciprocal of both sides of the equation to find the focal length (f):
f = 1400 / 81 ≈ 17.28 cm
The focal length of the diverging lens is approximately -17.28 cm (negative because it is a diverging lens).