An apple is placed 17.0 cm in front of a diverging lens with a focal length of magnitude 23.2 cm. What is the image distance i of the image of the apple through this lens?

What is the magnification of the image of the apple?
Wwhich of the following statements are true and which are false?
The image of the apple is:
behind the lens?
virtual?
inverted?
unchanged in size?

An entomologist examines a bug by holding a magnifying glass at arm’s length and 10.0 cm away from the bug. The magnifying glass has a focal length of 15.0 cm. Find the image distance and the magnification

Learn to use the lens (and mirror) equation:

1/Do + 1/Di = 1/f

The object distance is Do = 17.0 cm in your case.

The focal length is 23.2 cm in this case.

Solve for image distance Di

The magnification is |Di/Do|
(note absolute value signs|.

Di will be negative in this case. That means the image is virtual, and not inverted.

To determine the image distance and magnification of the image of the apple, we can use the lens formula and magnification formula.

1. Image distance (i):
The lens formula states:
1/f = 1/v - 1/u
Where f is the focal length of the lens, v is the image distance, and u is the object distance (distance of the apple from the lens).

Given:
focal length (f) = 23.2 cm
object distance (u) = -17.0 cm (negative sign indicates the object is placed in front of the lens)

Substituting the values into the lens formula and solving for v (image distance):
1/23.2 = 1/v - 1/-17
1/v = 1/23.2 + 1/17
v = 1/((1/23.2) + (1/17))
v ≈ 12.10 cm

Therefore, the image distance (i) of the apple is approximately 12.10 cm.

2. Magnification (m):
The magnification formula is given by:
m = -v/u
Where m is the magnification.

Substituting the values into the magnification formula:
m = -12.10/-17.0
m ≈ 0.71

Therefore, the magnification of the image of the apple is approximately 0.71.

Now, let's analyze the statements about the image:

- The image is behind the lens: False. Since the image distance (i) is positive (12.10 cm), it means the image is formed on the opposite side of the lens where the object is placed, which is in front of the lens.
- The image is virtual: True. The negative value of the object distance (u) indicates that the lens forms a virtual image of the object.
- The image is inverted: True. The negative magnification value (m) indicates that the image is formed inverted compared to the object.
- The image is unchanged in size: False. Since the magnification is not equal to 1, it means the image is either enlarged or reduced compared to the object. In this case, the magnification is approximately 0.71, indicating that the image is smaller than the object.

To find the image distance (i) of the apple through the diverging lens, we can use the lens equation:

1/f = 1/di - 1/do

Where:
f = focal length of the lens
di = image distance
do = object distance (distance of the apple from the lens)

In this case, the focal length (f) is given as 23.2 cm, and the object distance (do) is given as -17.0 cm (since the object is in front of the lens).

Using the lens equation, we can rearrange it to solve for di:

1/f = 1/di - 1/do
1/23.2 = 1/di - 1/-17.0

Simplifying the equation, we get:

1/23.2 = 1/di + 1/17.0

To solve for di, we can take the reciprocal of both sides:

23.2 = di/17.0 + 1
23.2 = di/17.0 + 17.0/17.0
23.2 = di/17.0 + 17.0/17.0
23.2 = di/17.0 + 1
23.2/17.0 = di/17.0 + 1

Simplifying further, we get:

di/17.0 = 23.2/17.0 - 1
di/17.0 = (23.2 - 17.0)/17.0
di/17.0 = 6.2/17.0

Finally, solving for di:

di = (6.2/17.0) * 17.0
di = 6.2 cm

Therefore, the image distance (i) of the apple through the lens is 6.2 cm.

To find the magnification (m) of the image of the apple, we can use the formula:

m = -di/do

Where:
di = image distance
do = object distance

Substituting the given values:

m = -6.2 / -17.0
m = 0.3647

Therefore, the magnification (m) of the image of the apple is approximately 0.3647.

Now for the true/false statements:

1. The image of the apple is behind the lens: True
Since the image distance (i) is positive (6.2 cm), it means the image is formed on the opposite side of the lens from the object, which is behind the lens.

2. The image of the apple is virtual: True
Since the image distance (i) is positive (6.2 cm), it means the image is formed on the same side of the lens as the object, which means it is virtual.

3. The image of the apple is inverted: True
Since the image is formed on the same side of the lens as the object and the magnification (m) is negative (-0.3647), it means the image is inverted.

4. The image of the apple is unchanged in size: False
Since the magnification (m) is not equal to 1, it means the image is not the same size as the object. In this case, the image is smaller than the object.