Question

A 1.80-m-tall person stands 3.50 m from a convex mirror and notices that he looks precisely half as tall as he does in a plane mirror placed at the same distance.

What is the radius of curvature of the convex mirror? (Assume that sinθ≈θ.) [Hint: The viewing angle is half.]

Answer 1

Step-by-step explanation:

Given:

height of the object
`h_o=1.90m`

height of the image
`h_i=(h_o)/(2)\\=h_i=(1.80)/(2)=0.90m`

object distance
`d_o=3.50m`

we know that:
`(h_i)/(h_o)=(-d_i)/(d_o)\\\\=d_i=(-h_i)/(h_o)* d_o\\\\=-(0.90)/(1.80)* (3.50)=-1.75m`

image
`d_i=-1.75m`

According to lens formular:

`(1)/(f)=(1)/(d_o)+(1)/(d_i)\\\\=(1)/(3.50)+-(1)/(1.75)\\\\f=-3.5`

focal length=
`(radius)/(2)\\\\radius=2* f\\=2* 3.5=7m`