If a man's face is 30.0 cm in front of a concave shaving mirror creating an upright image 1.50 times as large as the object,what is the mirror's focal length?
Answers
Given: the distance of man from the mirror, u = 30.0 cm
the height of the man, h₁ = x
the height of the image, h₂ = 1.50 x
To Find: the focal length of the mirror, f.
Solution:
To calculate f, the formula used:
- 1 / f = 1 / v + 1 / u ⇒ Mirror Formula
- Magnification of the mirror, m = - v / u
- or, m = h₂ / h₁
Applying the above formula for m:
m = -(- v) / -30
= - v / 30 ⇒ 1
m = 1.50x / x
The negative sign is used as per the sign convention rule for the concave mirror.
m = + 1.5 / 1 ⇒ 2
On equating equations 1 and 2:
- v / 30 = 1.5 / 1
-v = 1.5 x 30
= 4.5 x 10
= (45/10) x 10
= 45
v = - 45 cm
Now apply the formula for f:
1 / f = 1 / v + 1 / u
1 / f = 1 / (-45) + 1 / 30
= 1 / 30 - 1 / 45
= 1 / 90
1 / f = 1 / 90
Reciprocal on both the sides:
f = 90 cm
Hence the focal length of the mirror is 90 cm.