Chapter 17, Problem 87A

a.

To determine

The image position of a customer.

Answer to Problem 87A

  $1.5\text{m}$ behind the mirror

Explanation of Solution

Given:

The radius of curvature of the surveillance mirror, $R=-3.8\text{m}$ (since convex mirrors are used as surveillance mirrors)

The distance of customer from the mirror, $d_o=6.5\text{m}$

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:

The focal length of the convex mirror will be,

  $f=\frac{R}{2}=\frac{-3.8\text{m}}{2}=-1.9\text{}\text{m}$

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

  $\begin{array}{c}\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_o}\\ \frac{1}{-1.9}=\frac{1}{d_i}+\frac{1}{6.5}\\ d_i=-\left(\frac{6.5\times 1.9}{6.5+1.9}\right)\\ \approx -1.5\text{m}\end{array}$

Negative sign denotes that image is formed behind the mirror.

Conclusion:

Therefore, the image of customer is at a distance of $1.5\text{m}$ behind the mirror.

b.

To determine

The height of the image.

Answer to Problem 87A

  $0.4\text{m}$

Explanation of Solution

Given:

The radius of curvature of the surveillance mirror, $R=-3.8\text{m}$ (since convex mirrors are used as surveillance mirrors)

The distance of customer from the mirror, $d_o=6.5\text{m}$

From part (a), the image distance of the above customer, $d_i=-1.5\text{m}$

The height of the customer, $h_o=1.7\text{m}$

Formula used:

Magnification, $m=\frac{h_i}{h_o}=-\frac{d_i}{d_o}$

Where,

  $\begin{array}{l}{h}_i=\text{image height}\\ h_o=\text{object height}\end{array}$

Calculation:

Using the given and calculated values of object and image distance, the magnification produced by the mirror is,

  $\begin{array}{c}m=-\frac{d_i}{d_o}\\ =-\frac{\left(-1.5\right)}{6.5}\\ =0.23\end{array}$

Using the above and given values in magnification formula, the height of the image of the customer will be,

  $\begin{array}{c}m=\frac{h_i}{h_o}\\ 0.23=\frac{h_i}{1.7}\\ h_i\approx 0.4\text{m}\end{array}$

Conclusion:

Therefore, the height of the image of the customer who is $1.7\text{m}$ tall is $0.4\text{m}$ .