To find the image distance for an object placed in front of a concave mirror, we use the mirror equation:
[tex]\[ d_i = \frac{d_0 f}{d_0 - f} \][/tex]
where:
- [tex]\( d_0 \)[/tex] is the object distance,
- [tex]\( f \)[/tex] is the focal length,
- [tex]\( d_i \)[/tex] is the image distance.
Given values:
- [tex]\( d_0 = 30.0 \, \text{cm} \)[/tex],
- [tex]\( f = 15.0 \, \text{cm} \)[/tex].
Substitute these values into the mirror equation:
[tex]\[ d_i = \frac{30.0 \, \text{cm} \times 15.0 \, \text{cm}}{30.0 \, \text{cm} - 15.0 \, \text{cm}} \][/tex]
Calculate the numerator:
[tex]\[ 30.0 \, \text{cm} \times 15.0 \, \text{cm} = 450.0 \, \text{cm}^2 \][/tex]
Calculate the denominator:
[tex]\[ 30.0 \, \text{cm} - 15.0 \, \text{cm} = 15.0 \, \text{cm} \][/tex]
Now, divide the numerator by the denominator to find the image distance:
[tex]\[ d_i = \frac{450.0 \, \text{cm}^2}{15.0 \, \text{cm}} = 30.0 \, \text{cm} \][/tex]
Therefore, the image distance [tex]\( d_i \)[/tex] is:
[tex]\[ d_i = 30.0 \, \text{cm} \][/tex]
So, the correct answer is:
D. [tex]\( 30.0 \, \text{cm} \)[/tex]
