From a distance of 60 m, a photographer uses a telephoto lens (f = 300.0 mm) to take a picture of a charging rhinoceros. How far from the rhinoceros would the photographer have to be to record an image of the same size using a lens whose focal length is 45.0 mm?
To find the distance from the rhinoceros where the image size is the same, you can use the lens formula:
1/f = 1/v - 1/u
Where:
- f is the focal length of the lens
- v is the image distance (distance between the lens and the image)
- u is the object distance (distance between the lens and the object)
Given:
- For the telephoto lens, f = 300.0 mm, and the object distance u is 60 m.
To convert the distance 60 m to millimeters, multiply by 1000:
u = 60 m * 1000 mm/m = 60000 mm
Now, we can plug the values into the lens formula to find v:
1/300 = 1/v - 1/60000
To simplify the equation, we can put both fractions over a common denominator:
1/300 = (1 - 60000/v) / 60000
Now we can cross multiply and solve for v:
60000 = 300(1 - 60000/v)
Divide both sides by 300:
200 = 1 - 60000/v
Rearrange the equation to solve for v:
60000/v = 1 - 200
60000/v = -199
Multiply both sides by v:
60000 = -199v
Divide both sides by -199:
v = 60000 / -199
v ≈ -301.51 mm
Since the image distance cannot be negative, the value of v is not physically meaningful. This means that with a 45.0 mm focal length lens, it is not possible to record an image of the same size from a distance of 60 m.