A concave mirror has a radius of curvature of 30.0 cm. It is positioned so that the upright image of an object is 2.0 times the size of the object. How far is the object from the mirror?

A. 10.0 cm
B. 15.0 cm
C. 22.5 cm
D. 7.5 cm



Answer :

To solve this problem, we need to determine the distance of the object from a concave mirror given its radius of curvature and the magnification of the image.

1. Understand the given values:
- The radius of curvature [tex]\( R \)[/tex] of the concave mirror is 30.0 cm.
- The magnification [tex]\( m \)[/tex] of the image is 2.0 times the size of the object.

2. Determine the focal length [tex]\( f \)[/tex]:
- For a concave mirror, the focal length is half of the radius of curvature.
[tex]\[ f = \frac{R}{2} \][/tex]
Substituting the given radius of curvature:
[tex]\[ f = \frac{30.0 \, \text{cm}}{2} = 15.0 \, \text{cm} \][/tex]

3. Relate magnification to image distance [tex]\( d_i \)[/tex] and object distance [tex]\( d_o \)[/tex]:
- The magnification [tex]\( m \)[/tex] of a mirror is given by the ratio of the image distance to the object distance.
[tex]\[ m = \frac{d_i}{d_o} \][/tex]
Given that [tex]\( m = 2.0 \)[/tex]:
[tex]\[ \frac{d_i}{d_o} = 2.0 \implies d_i = 2 \cdot d_o \][/tex]

4. Use the mirror equation:
- The mirror equation relates the focal length to the object distance and the image distance.
[tex]\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \][/tex]
Substituting [tex]\( d_i = 2 \cdot d_o \)[/tex]:
[tex]\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{2 \cdot d_o} \][/tex]

5. Solve for [tex]\( d_o \)[/tex]:
- Combine the terms on the right side of the mirror equation:
[tex]\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{2 \cdot d_o} = \frac{2}{2 \cdot d_o} + \frac{1}{2 \cdot d_o} = \frac{3}{2 \cdot d_o} \][/tex]
- Invert both sides to solve for [tex]\( d_o \)[/tex]:
[tex]\[ \frac{2 \cdot d_o}{3} = f \][/tex]
- Substituting the given focal length [tex]\( f = 15.0 \, \text{cm} \)[/tex]:
[tex]\[ \frac{2 \cdot d_o}{3} = 15.0 \, \text{cm} \][/tex]
- Solve for [tex]\( d_o \)[/tex]:
[tex]\[ 2 \cdot d_o = 3 \cdot 15.0 \, \text{cm} = 45.0 \, \text{cm} \][/tex]
[tex]\[ d_o = \frac{45.0 \, \text{cm}}{2} = 22.5 \, \text{cm} \][/tex]

Therefore, the distance of the object from the mirror is:

[tex]\[ \boxed{22.5 \, \text{cm}} \][/tex]

The correct answer is C.