An image is created by a 2 cm focal length concave mirror. The image is located 3 cm from the mirror. How far from the mirror is the object?

Question

BlackTom AI · Accepted Answer

We start by identifying the given data: a concave mirror with focal length f = +2 cm (positive for a concave mirror in standard sign convention), and an image distance v = +3 cm (positive, indicating a real image on the same side as the object).

Apply the mirror formula: 1/f = 1/u + 1/v, where u is the object distance (negative in the usual Cartesian sign convention, but we will solve for its magnitude and interpret the result for the distance from the mirror).

Plug in the known values: 1/2 = 1/u + 1/3.

Compute 1/u: 1/u = 1/2 − 1/3 = (3 − 2)/6 = 1/6, so u = 6 cm (magnitude).

Interpretation: the object is located 6 cm from the mirror. Since the distance is a positive magnitude and the sign convention yields u = −6 cm in the standard Cartesian system, the distance from the mirror is 6 cm.

Now evaluate each option:

- Option a: 6 cm — This matches the calculated object distance. It is the correct magnitude of how far the object sits from the mirror, given the provided f and v.
- Option b: 1 cm — This would imply a much larger image or a very different combination of f and v; it does not satisfy the mirror equation with f = 2 cm and v = 3 cm.
- Option c: 8 cm — If u were 8 cm, the left side 1/u would be 1/8, and 1/f would need to equal 1/8 + 1/3 = 11/24, which would give f ≈ 2.18 cm, not the given f = 2 cm, so this is inconsistent.
- Option d: 5 cm — If u were 5 cm, 1/u = 1/5, and 1/f would be 1/5 + 1/3 = 8/15, so f would be 15/8 ≈ 1.875 cm, which again does not match the given focal length of 2 cm.

In summary, only the 6 cm distance satisfies the mirror equation with f = 2 cm and v = 3 cm, while all other distances fail to reconcile the given focal length or image distance.

An image is created by a 2 cm focal length concave mirror. The image is located 3 cm from the mirror. How far from the mirror is the object?

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