To determine which telescope has the greatest magnification, we need to use the magnification formula for telescopes. The magnification [tex]\( M \)[/tex] is given by the ratio of the focal length of the objective lens [tex]\( f_o \)[/tex] to the focal length of the eyepiece lens [tex]\( f_e \)[/tex]:
[tex]\[ M = \frac{f_o}{f_e} \][/tex]
Here's the step-by-step solution:
1. Identify the focal lengths of the objective lenses for each telescope:
- Telescope A: 5 cm
- Telescope B: 21 cm
- Telescope C: 37 cm
- Telescope D: 43 cm
2. Assume the focal length of the eyepiece lens is the same for all telescopes. Let's denote this common focal length as [tex]\( f_e \)[/tex].
3. Calculate the magnification for each telescope using the formula [tex]\( M = \frac{f_o}{f_e} \)[/tex]:
- For Telescope A:
[tex]\[
M_A = \frac{5}{f_e}
\][/tex]
- For Telescope B:
[tex]\[
M_B = \frac{21}{f_e}
\][/tex]
- For Telescope C:
[tex]\[
M_C = \frac{37}{f_e}
\][/tex]
- For Telescope D:
[tex]\[
M_D = \frac{43}{f_e}
\][/tex]
4. Compare the calculated magnifications:
- Telescope A: [tex]\( M_A = \frac{5}{f_e} = 5 \)[/tex]
- Telescope B: [tex]\( M_B = \frac{21}{f_e} = 21 \)[/tex]
- Telescope C: [tex]\( M_C = \frac{37}{f_e} = 37 \)[/tex]
- Telescope D: [tex]\( M_D = \frac{43}{f_e} = 43 \)[/tex]
5. Determine which telescope has the greatest magnification by comparing the numerical values:
- [tex]\( M_A = 5 \)[/tex]
- [tex]\( M_B = 21 \)[/tex]
- [tex]\( M_C = 37 \)[/tex]
- [tex]\( M_D = 43 \)[/tex]
Clearly, the magnification of Telescope D is the highest. Therefore, Telescope D would have the greatest magnification.
So the answer is:
D
