Radius of convex spherical mirror?

AI Thread Summary
The discussion centers on calculating the radius of a convex spherical mirror positioned 25 feet from a store's entrance, where the clerk needs to see a 6-foot person at least 3 inches tall in the mirror. Initial calculations incorrectly assumed the image was inverted, leading to a radius of 26 inches. After clarifying that the image produced by a convex mirror is virtual and upright, the correct radius was recalculated to be 24 inches. However, further analysis revealed that the radius of curvature should actually be 26 inches, aligning with the properties of convex mirrors. The final consensus confirms the radius of the mirror is indeed 26 inches.
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The problem:
"A convex spherical mirror is 25 ft from the door of a convenience store. The clerk needs to see a 6 ft. person entering the store at least 3 inches tall in the mirror to identify them. What is the radius of the mirror?"

d_obj = do = 25 ft = 300 inches
h_img = hi = 3 inches
h_obj = ho = 6 ft = 72 inches

The formulas I tried using:

focal point = f = radius/2 = r/2
1/do+1/di = 1/f = 1/r/2 = 2/r
For convex mirror: m = magnification = hi/ho = -di/do

Work:

hi/ho = -di/do
3/72 = -di/300
di=-12.5 inches

1/do + 1/di = 2/r
1/300 - 1/12.5 = 2/r
r=26 inches

I found the radius to be 26 inches, which seems a bit too large. Is it correct or am I doing something wrong? I feel as though I must somehow incorporate the magnification, but I'm not sure how. It would be -24X but looking at something normally you'll see it as smaller than it is if you're at a greater distance so I'm not sure how to compensate for that.
 
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The image is real and upside down, so the height of the image is negative. Therefore, di is positive.

ehild

Edit: I was wrong: The mirror is convex, so the image is upright (virtual). Di is positive!
 
Last edited:
ehild said:
The image is real and upside down, so the height of the image is negative. Therefore, di is positive.

ehild

d_obj = do = 25 ft = 300 inches
h_img = hi = -3 inches
h_obj = ho = 6 ft = 72 inches

hi/ho = -di/do
-3/72 = -di/300
di=12.5 inches

1/do + 1/di = 2/r
1/300 + 1/12.5 = 2/r
r=24 inches

So the radius of the mirror is precisely 24 inches? That still seems like a rather large convenience store mirror.
 
It is the radius of the curvature. A spherical mirror is a part of a spherical surface, its diameter can be as small as you like.

ehild
 
Okay, thank you!
 
Sorry I was wrong, mixed convex and concave. The clerk does not want to see people upside down! He/she uses a convex mirror, that produces virtual, upright image. For such an image, di is positive, but the focal distance is negative. From the mirror equation, you get f=-13 inch. So the 26 inch for the radius was correct.

ehild
 
Okay, thanks. I was wondering about the inverted image, though the result seemed better.
 
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