Detective Magnifying Glass Mystery: Solving for Focal Length
What is the focal length of the lens used for the magnifying glass held by the detective?
The magnification produced by a lens is given by the formula `M = v/u`, where `v` is the distance between the image and the lens, and `u` is the distance between the object and the lens. It can also be expressed as the ratio of the height of the image (`h'`) to the height of the object (`h`), which is given by `M = h'/h`.
In this case, the magnifying glass creates an upright image three times as large as the object, so `M = 3` and `h' = 3h`. The detective holds the magnifying glass 8.6 cm above the object he is studying, making `u = 8.6 cm`.
We need to find the focal length of the lens (`f`), which can be determined using the formula `1/f = 1/v + 1/u`. By substituting the values of `M` and `u` into the rearranged formula `v = Mu`, we can solve for the focal length.
Solving for the Focal Length:
Distance between object and lens (`u`) = 8.6 cm
Magnification (`M`) = 3
Steps to Find Focal Length:
1. Use the formula `v = Mu` to find the distance between image and lens (`v`):
Substituting the values of `M` and `u`, we get `v = 3 * 8.6 = 25.8 cm`
2. Now, use the formula `1/f = 1/v + 1/u` to find the focal length (`f`):
Substituting the values of `v` and `u`, we get `1/f = 1/25.8 + 1/8.6`
Simplifying further, `1/f = 0.0388 + 0.1163`
Combining the fractions, `1/f = 0.1551`
Therefore, the focal length of the lens used for the magnifying glass is `f = 1/0.1551 ≈ 6.45 cm`
Conclusion:
The focal length of the lens used for the magnifying glass held by the detective is approximately 6.45 cm.