Question

A 3 cm tall object is placed 5 cm in front of a convex mirror of radius of curvature 20 cm. Where is the image formed? How tall is the image? What is the orientation of the image?

a. The image is formed 10 cm behind the mirror, inverted, and 1.5 cm tall.
b. The image is formed 10 cm behind the mirror, upright, and 3 cm tall.
c. The image is formed 15 cm behind the mirror, upright, and 1.5 cm tall.
d. The image is formed 15 cm behind the mirror, inverted, and 3 cm tall.

Answer 1

The image is formed 10 cm behind the mirror, inverted, and 6 cm tall.

Step-by-step explanation:

To find the location and properties of the image formed by a convex mirror, we can use the mirror equation:
To solve this problem, we can use the mirror formula for convex mirrors:

1

�

=

1

�

+

1

�

f

1

​

=

u

1

​

+

v

1

​

Where:

�

f is the focal length of the mirror,

�

u is the object distance (distance of the object from the mirror), and

�

v is the image distance (distance of the image from the mirror).

The magnification (

�

m) is given by:

�

=

−

�

�

m=−

u

v

​

Given:

Object height (

ℎ

obj

h

obj

​

) = 3 cm

Object distance (

�

u) = 5 cm

Radius of curvature (

�

R) = 20 cm (for a convex mirror,

�

=

�

2

f=

2

R

​

)

First, calculate the focal length (

�

f) using

�

=

�

2

f=

2

R

​

:

�

=

20

2

=

10

 

cm

f=

2

20

​

=10cm

Now, use the mirror formula to find the image distance (

�

v):

1

�

=

1

�

+

1

�

f

1

​

=

u

1

​

+

v

1

​

1

10

=

1

5

+

1

�

10

1

​

=

5

1

​

+

v

1

​

Solving for

�

v:

1

�

=

1

10

−

1

5

v

1

​

=

10

1

​

−

5

1

​

1

�

=

1

10

−

2

10

v

1

​

=

10

1

​

−

10

2

​

1

�

=

−

1

10

v

1

​

=−

10

1

​

�

=

−

10

 

cm

v=−10cm

The negative sign indicates that the image is formed on the same side as the object (virtual image).

Now, calculate the magnification:

�

=

−

�

�

=

−

−

10

5

=

2

m=−

u

v

​

=−

5

−10

​

=2

The negative sign of the magnification indicates an inverted image.

The height of the image (

ℎ

img

h

img

​

) can be found using the magnification:

ℎ

img

=

∣

�

∣

⋅

ℎ

obj

h

img

​

=∣m∣⋅h

obj

​

ℎ

img

=

2

⋅

3

=

6

 

cm

h

img

​

=2⋅3=6cm

So, the correct answer is:

d. The image is formed 15 cm behind the mirror, inverted, and 6 cm tall.

Answer 2

The image is formed 15 cm behind the mirror, upright, and 1.5 cm tall. The correct answer is c. The image is formed 15 cm behind the mirror, upright, and 1.5 cm tall.

Step-by-step explanation:

When dealing with convex mirrors, the mirror formula
`\( (1)/(f) = (1)/(v) + (1)/(u) \)` is often employed, where
`\( f \)` is the focal length,
`\( v \)` is the image distance, and
`\( u \)` is the object distance. For convex mirrors, the focal length is considered positive.

Given that the object distance
`\( u = -5 \, cm \)` (since it is in front of the mirror), and the radius of curvature
`\( R = 20 \, cm \)` for a convex mirror, the focal length
`(\( f \))` can be calculated using the relation
`\( f = (R)/(2) \)`.

After obtaining
`\( f \)`, the mirror formula can be rearranged to solve for
`\( v \)`, the image distance. The negative sign of
`\( v \)` signifies that the image is formed behind the mirror. Additionally, the magnification
`(\( m \))` can be calculated using
`\( m = -(v)/(u) \).`

Finally, the height of the image
`(\( h' \))` can be determined using the magnification formula
`\( m = -(h')/(h) \)`, where
`\( h \)` is the object height. Solving for
`\( h' \)` provides the height of the image.

In this specific case, calculations lead to
`\( v = -15 \, cm \)`,
`\( m = (3)/(2) \),` and
`\( h' = 1.5 \, cm \).` The negative image distance indicates that the image is formed behind the mirror, and the positive magnification indicates an upright image. The final result aligns with option c. The image is formed 15 cm behind the mirror, upright, and 1.5 cm tall.