Certainly!
To solve the problem of finding the radius of curvature of the convex mirror given the positions of the object and the resulting image system, we will use the mirror formula and the relation between the focal length and the radius of curvature.
### Step-by-Step Solution:
1. Determine the object and image distances:
- Let the object distance be [tex]\( u \)[/tex]. According to the problem, the object is virtual and placed at a distance of 15 cm from the mirror.
[tex]\[
u = -15 \, \text{cm} \quad (\text{object distance is considered negative for virtual objects in convex mirrors})
\][/tex]
- Let the image distance be [tex]\( v \)[/tex]. The image is formed at a distance of 5 cm from the mirror.
[tex]\[
v = 5 \, \text{cm} \quad (\text{image distance is positive for a virtual image in convex mirrors})
\][/tex]
2. Use the mirror formula:
The mirror formula, which relates the object distance [tex]\( u \)[/tex], the image distance [tex]\( v \)[/tex], and the focal length [tex]\( f \)[/tex] of the mirror, is given by:
[tex]\[
\frac{1}{f} = \frac{1}{v} + \frac{1}{u}
\][/tex]
3. Substitute the given values into the formula:
[tex]\[
\frac{1}{f} = \frac{1}{5 \, \text{cm}} + \frac{1}{-15 \, \text{cm}}
\][/tex]
[tex]\[
\frac{1}{f} = \frac{1}{5} - \frac{1}{15}
\][/tex]
4. Calculate the focal length [tex]\( f \)[/tex]:
Find a common denominator for the fractions:
[tex]\[
\frac{1}{f} = \frac{3}{15} - \frac{1}{15} = \frac{2}{15}
\][/tex]
Therefore,
[tex]\[
f = \frac{15}{2} \, \text{cm} \approx 7.5 \, \text{cm}
\][/tex]
(The small rounding difference depends on the exact arithmetic precision used.)
5. Calculate the Radius of Curvature:
The radius of curvature [tex]\( R \)[/tex] is related to the focal length [tex]\( f \)[/tex] by the formula:
[tex]\[
R = 2f
\][/tex]
Substitute the focal length we just calculated:
[tex]\[
R = 2 \times 7.5 \, \text{cm} \approx 15 \, \text{cm}
\][/tex]
(Similarly, specific numerical outputs can slightly adjust this value.)
### Summary:
- The focal length [tex]\( f \)[/tex] of the convex mirror is approximately [tex]\( 7.5 \, \text{cm} \)[/tex].
- The radius of curvature [tex]\( R \)[/tex] of the convex mirror is approximately [tex]\( 15 \, \text{cm} \)[/tex].
So the radius of curvature is 15 cm, and the focal length is 7.5 cm.
