A large convex spherical mirror in an amusement park is facing a plane mirror 10 m away. A boy of height 1 m standing midway between the two sees himself twice as tall as in plane mirror as in spherical one. In other words, the angle subtended at the observer by the image in plane mirror is twice the angle subtended by the image in the spherical mirror. What is the focal length of the convex mirror?

Engineering

Physics

Spherical Mirror

Question

A large convex spherical mirror in an amusement park is facing a plane mirror 10 m away. A boy of height 1 m standing midway between the two sees himself twice as tall as in plane mirror as in spherical one. In other words, the angle subtended at the observer by the image in plane mirror is twice the angle subtended by the image in the spherical mirror. What is the focal length of the convex mirror?

A

5 m

B

2.5 m

C

1.8 m

D

3.6 m

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$\frac{1}{10} = θ_{1} = 0 .1 rad .$

$\frac{h_{I}}{1} = \frac{5 f}{\frac{5 + f}{5}}$

$h_{I} = \frac{f}{5 + f}$

$θ_{2} = \frac{\frac{f}{5 + f}}{5 + \frac{5 f}{5 + f}} = \frac{f}{25 + 10 f} = \frac{0 .1}{2}$

2f = 2.5 + f

f = 2.5 m

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