Chapter 17, Problem 58A

To determine

The image distance from the mirror.

Answer to Problem 58A

75 cm in front of a mirror

Explanation of Solution

Given:

The radius of curvature of the concave mirror, $R=150\text{cm}$

The object distance from the mirror, $d_o=\infty$ (since the position of star will be at infinite distance as compared to the radius of curvature of the mirror)

Formula used:

The relation between the radius of curvature of mirror $\left(R\right)$ and its focal length $\left(f\right)$ is,

  $f=\frac{R}{2}$

Mirror equation,

  $\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_o}$

Where,

  $\begin{array}{l}f=\text{focal length}\\ d_i=\text{ image distance}\\ d_o=\text{object distance}\end{array}$

Calculation:

Then, the focal length of the concave mirror will be,

  $f=\frac{R}{2}=\frac{150\text{}\text{}\text{cm}}{2}=75\text{}\text{cm}$

Using the given values in Mirror equation, the image distance will be,

  $\begin{array}{c}\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_o}\\ \frac{1}{75}=\frac{1}{d_i}+\frac{1}{\infty }\\ \frac{1}{75}=\frac{1}{d_i}+0\\ d_i=75\text{cm}\end{array}$

Conclusion:

Hence, the image will be formed at a distance of $75\text{cm}$ in front of a mirror.