A concave shaving mirror has a radius of curvature of 60\text{ cm}. A person places their … — JAMB UTME

A concave shaving mirror has a radius of curvature of $60\text{ cm}$ . A person places their face in front of the mirror such that an upright image magnified $3$ times is formed. What is the distance of the face from the mirror, in centimeters?

Cevap: 20 cm

Cevap

The distance of the person's face from the mirror is

20\text{ cm}

For a concave mirror with a radius of curvature of

60\text{ cm}

, the focal length is

f = +30\text{ cm}

. An upright image is virtual, corresponding to a positive magnification

m = +3

. Since

m = -\frac{v}{u}

, the image distance is

v = -3u

. Substituting these into the mirror formula

\frac{1}{f} = \frac{1}{u} + \frac{1}{v}

gives

\frac{1}{30} = \frac{1}{u} - \frac{1}{3u} = \frac{2}{3u}

, solving to

u = 20\text{ cm}

Adım Adım Çözüm

Determine the focal length of the concave mirror.

f = 30\text{ cm}

The focal length is half the radius of curvature (

f = \frac{r}{2} = \frac{60\text{ cm}}{2} = 30\text{ cm}

Express the image distance

v

in terms of the object distance

u

using the magnification relationship.

v = -3u

An upright image produced by a spherical mirror is virtual, so linear magnification

m = +3

. Using

m = -\frac{v}{u} = +3

, we obtain

v = -3u

Substitute

f

and

v

into the mirror equation to solve for

u

u = 20\text{ cm}

Applying the mirror formula

\frac{1}{f} = \frac{1}{u} + \frac{1}{v} \implies \frac{1}{30} = \frac{1}{u} - \frac{1}{3u} = \frac{2}{3u} \implies 3u = 60 \implies u = 20\text{ cm}

Anahtar Kavram

Mirror equation and sign conventions for virtual images formed by concave mirrors

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