Where should the birthday candle of height 5 cm be placed in front of a convex lens of focal length 15 cm so that a real image will be formed at a distance of 25 cm from the lens? What is the height of this image?

To determine where the birthday candle should be placed and the height of the resulting image, we can apply the lens formula:

1/f = 1/v - 1/u

where:
f is the focal length of the lens,
v is the distance of the image from the lens, and
u is the distance of the object from the lens.

In this case, the focal length (f) is given as 15 cm, and the distance of the image (v) is given as 25 cm. We need to find the distance of the object (u) and the height of the image.

Let's begin by rearranging the lens formula to solve for u:

1/u = 1/f - 1/v

Substituting the given values:

1/u = 1/15 - 1/25

Finding the common denominator:

1/u = (5 - 3)/75

Simplifying:

1/u = 2/75

To find u, we take the reciprocal on both sides:

u = 75/2
u = 37.5 cm

Now that we have the distance of the object (u), we can calculate the height of the image using the magnification formula:

magnification (m) = height of the image (h') / height of the object (h)

The magnification can be expressed as:

m = -v/u

Substituting the given values:

m = -25/37.5

Simplifying:

m = -2/3

Since the height of the object (h) is given as 5 cm, we can calculate the height of the image (h'):

m = h'/h

-h' / 5 = -2/3

Cross-multiplying:

3h' = 10

Dividing both sides by 3:

h' = 3.33 cm

Therefore, the birthday candle should be placed at a distance of 37.5 cm from the convex lens for a real image to be formed at 25 cm. The resulting image will have a height of approximately 3.33 cm.