An object that is 3.4 mm tall is placed 25 cm from the vertex of a convex spherical mirror. The radius of curvature of the mirror is 73 cm. What is the height of the image?

Question

BlackTom AI · Accepted Answer

We begin by identifying the given quantities and the sign conventions for a convex mirror.
- Object height h_o = 3.4 mm
- Object distance s = 25 cm (positive in front of the mirror)
- Radius of curvature R = 73 cm for a convex mirror. For mirrors, the focal length f is defined as f = R/2 with a sign: for a convex (diverging) mirror, f is negative. Thus f = -R/2 = -36.5 cm.

Next, apply the mirror equation to find the image distance s'.
1/f = 1/s + 1/s'  →  1/s' = 1/f − 1/s
Plugging in the values: 1/s' = (−1/36.5) − (1/25) ≈ −0.027397 − 0.04 ≈ −0.067397.
s' ≈ −14.83 cm (negative indicates a virtual image behind the mirror).

Then determine the magnification m, which relates image height h_i to object height h_o via m = h_i/h_o = −s'/s.
m = −(−14.83)/25 ≈ 14.83/25 ≈ 0.593.

Now find the image height: h_i = m × h_o = 0.593 × 3.4 mm ≈ 2.02 mm.

Option-by-option assessment:
- 3.4 mm: This would imply m = 1.0 (image same size as object). The calculated magnification is about 0.593, so this option is inconsistent with the derived geometry and sign conventions.
- 6.8 mm: This corresponds to a magnification of about m ≈ 2.0 (image larger than the object). For a convex mirror with the given distances, the image is virtual and diminished, not enlarged, so this is incorrect.
- 2.0 mm: This matches the computed image height when using the proper sign convention for f and s, yielding m ≈ 0.593 and h_i ≈ 2.02 mm. This is the consistent result.
- 1.0 mm: This would require a magnification of about m ≈ 0.294. That would imply a different s' that does not satisfy the mirror equation with f = −36.5 cm and s = 25 cm, so it does not align with the given data.

In summary, applying the mirror equation with f = −36.5 cm and s = 25 cm leads to a virtual, diminished image with height about 2.0 mm, which aligns with the standard behavior of a convex mirror.

An object that is 3.4 mm tall is placed 25 cm from the vertex of a convex spherical mirror. The radius of curvature of the mirror is 73 cm. What is the height of the image?Single choice

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