A dedicated sports car enthusiast polishes the inside and outsidesurfaces of a h

Question

A dedicated sports car enthusiast polishes the inside and outsidesurfaces of a hubcap that is a section of a sphere. When he looksinto one side of the hubcap, he sees an image of hisface 26.0 cm in back of thehubcap. He then turns the hubcap over, keeping it the same distancefrom his face. He now sees an image of his face 10.0cm in back of the hubcap. (a) How far is his face from the hubcap?
cm
(b) What is the radius of curvature of the hubcap?
cm
(a) How far is his face from the hubcap?
cm
(b) What is the radius of curvature of the hubcap?
cm

Explanation / Answer

   let us take that the magnitude of the radius ofcurvature |R| is same for both sides of the hubcap
   for the convex side R = - |R|
   for the concave side R = + |R|
   the object distance p is positive (real object) andhas the same value in both the cases
   the virtual image distance as q = - |q|
   so from the mirror equation we get
   for the convex side as
   (1 / - |q|) = (2 / - |R|) - (1 / p)
   |q| = |R| p / |R| + 2 p .......... (1)
   for the concave side as
   (1 / - |q|) = (2 / |R|) - (1 / p)
   |q| = |R| p / |R| - 2 p .......... (2)
   on comparing equations (1) and (2) we can see that thesmaller magnitude image distance
   |q| = 10.0 cm occurs with the convex side of themirror so we get
   (1 / - 10.0 cm) = (2 / - |R|) - (1 / p)........(3)
   for the concave side |q| = 26.0 cm gives
   (1 / - 26 cm) = (2 / |R|) - (1 / p) ........(4)
(a)
   adding equations (3) and (4) we get
   p = ....... cm
(b)
   subtracting (3) from (4) we get
   |R| = ........ cm

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