Chapter 17, Problem 4STP

To determine

To Choose: The correct option.

Answer to Problem 4STP

D. $19\text{cm}$

Explanation of Solution

Given:

The focal length of the concave mirror, $f=24.0\text{cm}$

The height of the image, $h_i=-8.5\text{cm}$ (negative sign denotes inverted image)

The image distance from the mirror, $d_i=34.5\text{cm}$

Formula used:

Mirror equation,

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

Where,

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

And,

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

Where,

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

Calculation:

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

  $\begin{array}{c}\frac{1}{f}=\frac{1}{d_i}+\frac{1}{d_o}\\ \frac{1}{24}=\frac{1}{34.5}+\frac{1}{d_o}\\ d_o=\frac{34.5\times 24}{34.5-24}\\ =78.85\text{cm}\end{array}$

Then, the object height will be,

  $\begin{array}{c}\frac{h_i}{h_o}=-\frac{d_i}{d_o}\\ \frac{-8.5}{h_o}=-\frac{34.5}{78.85}\\ h_o\approx 19\text{cm}\end{array}$

Conclusion:

Hence, the object’s height is $19\text{cm}$ .