A converging lens (f = 27.5 cm) is used to project an image of an object onto a screen. The object and the screen are 113 cm apart, and between them the lens can be placed at either of two locations. Find the two object distances. (Enter your answers from smallest to largest.)
I have tried this problem, but I don't even know where to begin, I have tried just splitting the distance between the two and using that, but nothing seems to work, thanks for any help!
To solve this problem, we can use the lens equation:
1/f = 1/di + 1/do
Where:
- f is the focal length of the lens
- di is the image distance (distance between the lens and the screen)
- do is the object distance (distance between the lens and the object)
We are given:
- f = 27.5 cm
- di = 113 cm
We need to find the two possible values for do.
To start, let's find the first value of do:
1/f = 1/di + 1/do
Rearranging the equation:
1/do = 1/f - 1/di
Substituting the given values:
1/do = 1/27.5 - 1/113
Now, let's compute this value:
1/do = (113 - 27.5) / (27.5 * 113)
Now, we can find the first value of do:
do = 1 / ( (113 - 27.5) / (27.5 * 113) )
Carrying out the calculation gives:
do ≈ 38.39 cm
Now, let's find the second value of do:
1/do = -1/f - 1/di
Substituting the values:
1/do = -1/27.5 - 1/113
Now, compute this value:
1/do = (27.5 + 113) / (27.5 * 113)
Now, we can find the second value of do:
do = 1 / ( (27.5 + 113) / (27.5 * 113) )
Carrying out the calculation gives:
do ≈ 3.53 cm
So, the two possible object distances are approximately 3.53 cm and 38.39 cm, from smallest to largest.