Answer:
To find the distance of the image formed by the convex mirror, we need to use the mirror equation:
\[ \frac{1}{f} = \frac{1}{d_o} + \frac{1}{d_i} \]
Where:
- \( f \) is the focal length of the mirror.
- \( d_o \) is the distance of the object from the mirror (object distance).
- \( d_i \) is the distance of the image from the mirror (image distance).
Given the radius of curvature \( R \) of the convex mirror is 10 cm, we can find the focal length \( f \):
\[ f = \frac{R}{2} \]
\[ f = \frac{10}{2} = 5 \, \text{cm} \]
The focal length of a convex mirror is positive. So, \( f = 5 \, \text{cm} \).
The object distance \( d_o \) is given as 15 cm (the object is placed in front of the mirror, so it is positive).
Now, use the mirror equation to find the image distance \( d_i \):
\[ \frac{1}{5} = \frac{1}{15} + \frac{1}{d_i} \]
First, solve the equation:
\[ \frac{1}{d_i} = \frac{1}{5} - \frac{1}{15} \]
\[ \frac{1}{d_i} = \frac{3}{15} - \frac{1}{15} \]
\[ \frac{1}{d_i} = \frac{2}{15} \]
Now, solve for \( d_i \):
\[ d_i = \frac{15}{2} = 7.5 \, \text{cm} \]
Therefore, the image is formed at a distance of 7.5 cm behind the convex mirror
