The owner of a van installs a rear-window lens that has a focal length of -0.320 m. When the owner looks out through the lens at an object located directly behind the van, the object appears to be 0.240 m from the back of the van, and appears to be 0.330 m tall. (a) How far from the van is the object actually located, and (b) how tall is the object?

To answer these questions, we can use the lens formula:

1/f = 1/v - 1/u

Where:
f is the focal length of the lens (given as -0.320 m, negative because it's a diverging lens),
v is the apparent distance of the object from the lens (0.240 m),
and u is the actual distance of the object from the lens (what we need to find).

(a) Finding the actual distance of the object from the van (u):

1/f = 1/v - 1/u

Rearranging the formula, we can solve for u:

1/u = 1/v - 1/f

Substituting the given values:

1/u = 1/0.240 - 1/-0.320

Simplifying:

1/u = 4.167 - (-3.125)

1/u = 7.292

Taking the reciprocal of both sides:

u = 1/7.292

u = 0.137 m

Therefore, the actual distance of the object from the van is approximately 0.137 m.

(b) Finding the height of the object:

To find the height of the object, we can use the magnification formula:

magnification (m) = -v/u

Substituting the given values:

m = -(0.240)/(0.137)

Simplifying:

m = -1.752

The negative sign indicates an inverted image.

The height of the object (h) is related to its apparent height (h') by the magnification formula:

m = h'/h

Substituting the magnification value:

-1.752 = h'/h

Rearranging the formula to solve for h:

h = h' / -1.752

Substituting h' = 0.330 m:

h = 0.330 / -1.752

h ≈ -0.188 m

Since the height of an object cannot be negative, we take the absolute value:

The object's height is approximately 0.188 m.

Therefore, the actual distance of the object from the van is approximately 0.137 m, and the height of the object is approximately 0.188 m.