Question

A convex mirror of focal length 34 cm forms an image of a soda bottle at a distance of 18 cm behind the mirror. The height of the image is 6.8 cm. a) Where is the object located? Answer in units of cm.
b) What is the magnification of the image?
c) Describe the image.
1. real, inverted, larger
2. real, upright, larger
3. virtual, inverted, larger 4. virtual, upright, larger 5. real, inverted, smaller
6. None of these
7. real, upright, smaller
8. virtual, inverted, smaller 9. virtual, upright, smaller

d) How tall is the object? Answer in units of cm.

Answer 1

Answers:

In convex mirrors the focus is virtual and the focal distance is negative. This is how the reflected rays diverge and only their extensions are cut at a point on the main axis, resulting in a virtual image of the real object .

a) Position of the soda bottle (the object)

The Mirror equation is:

$\frac{1}{f}=\frac{1}{u}+\frac{1}{v}$   (1)

Where:

$f$ is the focal distance

$u$ is the distance between the object and the mirror  

$v$ is the distance between the image and the mirror

 

We already know the values of $f$ and $v$, let's find $u$ from (1):

$u=\frac{v.f}{v-f}$   (2)

Taking into account the explanation at the beginign of this asnswer:

$f=-34cm$ and $v=-18cm$

The negative signs indicate the focal distance and the distance between the image and the mirror are virtual

Then:

$u=\frac{(-18cm)(-34cm)}{-18cm-(-34cm)}$  

$u=38.25cm$ (3) >>>>Position of the soda bottle

b) magnification of the image

The magnification $m$ of the image is given by:

$m=-\frac{v}{u}$ (4)

$m=-\frac{(-18cm)}{38.25cm}$  

$m=0.47$ (5)>>>the image is 0.47 smaller than the object

The fact that this value is positive means the image is upright

c) Describe the image

According to the explanations and results obtained in the prior answers, the correct option is 9:

The image is virtual, upright, smaller than the object

d) Height of the soda bottle (object)

Another way to find the magnification is by the following formula:

$m=\frac{h_{i}}{h_{o}}$ (6)

Where:

$h_{i}$ is the image height

$h_{o}$ is the object height

We already know the values of $m$ and $h_{i}$, let's find $h_{o}$:

$h_{o}=\frac{h_{i}}{m}$ (7)

$h_{o}=\frac{6.8cm}{0.47}$  

$h_{o}=14.46cm$ (8) >>>height of thesoda bottle