A concave makeup mirror is designed so that a person 29 cm in front of it sees an upright image magnified by a factor of two. What is the radius of curvature of the mirror?
To find the radius of curvature of the concave mirror, we can use the mirror formula, which relates the object distance (distance from the object to the mirror), image distance (distance from the image to the mirror), and the radius of curvature.
The mirror formula is given as:
1/f = 1/d_o + 1/d_i
Where:
f = focal length of the mirror
d_o = object distance
d_i = image distance
In this case, we are given that the person is 29 cm in front of the mirror, which means the object distance (d_o) is 29 cm. And we are also given that the person sees an upright image magnified by a factor of two, which means the image distance (d_i) is somewhere behind the mirror.
From the given information, we can deduce that the magnification (m) is 2. Since the image is upright and magnified, the magnification value is positive.
m = -d_i/d_o
By substituting the given values, we have:
2 = -d_i/29
Solving for d_i, we get:
d_i = -58 cm
Now, substitute the values of d_o and d_i into the mirror formula, and solve for f:
1/f = 1/29 + 1/-58
This simplifies to:
1/f = -2/58
Simplifying further:
1/f = -1/29
Now, solving for f:
f = -29 cm
Since the focal length (f) of a concave mirror is half of its radius of curvature (R), we have:
R = -2f
Substituting the value of f, we get:
R = -2 * (-29) cm
R = 58 cm
Therefore, the radius of curvature of the concave mirror is 58 cm.