An amusement park mirror shows anyone who stands 2.5 meters in front of it an upright image two times the person's height. What is the mirror's radius of curvature?
To find the mirror's radius of curvature, we can make use of the mirror equation:
1/f = 1/d_o + 1/d_i,
where f is the focal length of the mirror, d_o is the object distance (distance from the mirror to the object), and d_i is the image distance (distance from the mirror to the image).
In this case, the mirror is creating an upright image that is twice the height of the person. So, the image distance, d_i, is equal to 2 times the object distance, d_o.
Given that the person is standing 2.5 meters in front of the mirror, we can denote the object distance as d_o = 2.5 meters.
Substituting these values into the mirror equation, we get:
1/f = 1/2.5 + 1/(2 × 2.5).
Simplifying this equation, we have:
1/f = 1/2.5 + 1/5.
Now, let's find the common denominator:
1/f = 2/5 + 1/5.
1/f = 3/5.
To isolate f, take the reciprocal of both sides:
f = 5/3.
Therefore, the mirror's radius of curvature is 5/3 meters, or approximately 1.67 meters.