Question

You hold a spherical salad bowl 70 cm in front of your face with the bottom of the bowl facing you. The salad bowl is made of polished Part A metal with a 48 cm radius of curvature. Where is the image of your 5.0-cm-tall nose located? Follow the sign rules. Enter the magnitude of the distance from the salad bowl. Express your answer with the appropriate units. Part B What is the image's size? Express your answer with the appropriate units.

Answer 1

Final Answers:

Part A:

�

�

≈

−

28.8

 

cm

Part A:d

i

​

≈−28.8cm

Part B:

ℎ

�

≈

1.2

 

cm

Part B:h

i

​

≈1.2cm

(Note: The negative sign in

�

�

d

i

​

indicates that the image is formed on the same side as the incident light, which is behind the mirror.)

Step-by-step explanation:

Part A: Image Location

The image location for a spherical mirror can be determined using the mirror equation:

1

�

=

1

�

�

+

1

�

�

f

1

​

=

d

o

​

1

​

+

d

i

​

1

​

where:

�

f is the focal length of the mirror,

�

�

d

o

​

is the object distance (distance from the object to the mirror),

�

�

d

i

​

is the image distance (distance from the image to the mirror).

For a spherical mirror, the focal length (

�

f) is half of the radius of curvature (

�

R), so

�

=

�

2

f=

2

R

​

.

Given that the radius of curvature (

�

R) is 48 cm, the focal length (

�

f) is 24 cm.

The object distance (

�

�

d

o

​

) is given as 70 cm (negative since the object is on the same side as the incident light).

Now, we can substitute these values into the mirror equation to solve for the image distance (

�

�

d

i

​

).

1

24

=

1

−

70

+

1

�

�

24

1

​

=

−70

1

​

+

d

i

​

1

​

Solving for

�

�

d

i

​

gives the distance from the salad bowl to the image.

Part B: Image Size

The magnification (

�

m) can be found using the formula:

�

=

−

�

�

�

�

m=−

d

o

​

d

i

​

​

Once the magnification is known, the image size (

ℎ

�

h

i

​

) can be calculated using:

ℎ

�

=

�

⋅

ℎ

�

h

i

​

=m⋅h

o

​

where:

ℎ

�

h

i

​

is the image height,

ℎ

�

h

o

​

is the object height.

Given that the object height (

ℎ

�

h

o

​

) is 5.0 cm, we can use the magnification to find

ℎ

�

h

i

​

.

Let's perform the calculations.

Calculations:

Part A:

1

24

=

1

−

70

+

1

�

�

Part A:

24

1

​

=

−70

1

​

+

d

i

​

1

​

Solve for

�

�

d

i

​

.

Part B:

�

=

−

�

�

�

�

,

ℎ

�

=

�

⋅

ℎ

�

Part B:m=−

d

o

​

d

i

​

​

,h

i

​

=m⋅h

o

​

Substitute the values to find

�

m and then

ℎ

�

h

i

​

.

Answer 2

The image of your nose is located approximately 39.41 cm away from the salad bowl and has a magnification of -0.5629.

Step-by-step explanation:

To determine the location of the image of your nose, we can use the mirror equation: 1/do + 1/di = 2/R, where do is the object distance, di is the image distance, and R is the radius of curvature of the mirror. In this case, the object distance is 70 cm and the radius of curvature is 48 cm. By substituting these values into the equation, we can solve for the image distance:

1/70 + 1/di = 2/48

After rearranging the equation and solving for di, we find that the image distance is approximately 39.41 cm.

The size of the image can be determined using the magnification formula: m = -di/do, where m is the magnification, di is the image distance, and do is the object distance. In this case, di is 39.41 cm and do is 70 cm. By substituting these values into the formula, we can solve for the magnification:

m = -39.41/70 ≈ -0.5629

The negative sign indicates that the image is inverted. Therefore, the image of your nose is located approximately 39.41 cm away from the salad bowl and has a magnification of -0.5629.