A slide is 5 inches high. it is placed 5.5 inches from the convex lens of a porjection lantern. The image is cast upon a screen 55 inches away. the focal length of the lens is ______inches?

b) the heigh of the image cast by the lens is ________inches?

(a) Use the lens formula.

1/5.5 + 1/55 = 1/f
Solve for the focal length, f.
10/55 + 1/55 = 11/55 = 1/5 = 1/f
One more step.

(b) image height = (object height)x(55/5.5)
= 5*10 = ___ inches

To determine the focal length of the lens, we can use the lens formula:

1/f = 1/d₀ + 1/dᵢ

where:
f is the focal length of the lens
d₀ is the object distance (distance of the slide from the lens)
dᵢ is the image distance (distance of the screen from the lens)

In this case, we are given:
d₀ = 5.5 inches
dᵢ = 55 inches

Substituting these values into the lens formula, we get:

1/f = 1/5.5 + 1/55

Simplifying the equation:

1/f = (55 + 5.5) / (5.5 * 55)

1/f = 60.5 / 302.5

1/f = 0.2

To find the focal length, we can take the reciprocal of both sides:

f = 1 / 0.2
f = 5 inches

Therefore, the focal length of the lens is 5 inches.

Now, let's determine the height of the image cast by the lens. To do this, we can use the magnification formula:

magnification (m) = -dᵢ / d₀

In this case, we are given:
d₀ = 5.5 inches
dᵢ = 55 inches

Substituting these values into the magnification formula, we get:

m = -55 / 5.5

Simplifying the equation:

m = -10

The negative sign indicates that the image is inverted.

Now, we can find the height of the image by multiplying the magnification by the height of the slide:

height of the image = m * height of the slide
height of the image = -10 * 5 inches
height of the image = -50 inches

Therefore, the height of the image cast by the lens is -50 inches.