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"Okay, so I need to find the diameter of the image of a planet at the focus of the Magellan telescope. The telescope has a diameter of 6.5 meters and an effective focal length of 72 meters. The angular diameter of the planet is given as 45 arcseconds. Hmm, let me think about how to approach this.\n\nFirst, I rememb... | [
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1... | 1.6 | <|im_start|>user
Each of the two Magellan telescopes has a diameter of $6.5 \mathrm{~m}$. In one configuration the effective focal length is $72 \mathrm{~m}$. Find the diameter of the image of a planet (in $\mathrm{cm}$ ) at this focus if the angular diameter of the planet at the time of the observation is $45^{\prime \prime}$.
Please reason step by step, and put your final answer within \boxed{}.<|im_end|>
<|im_start|>assistant
<think>
| [
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... | sharding_1_2025-07-16 23:36.jsonl | minerva_math | 20,973 |
["Okay, so I need to compute the luminosity of a white dwarf star. The problem gives me the effectiv(...TRUNCATED) | [151644,872,198,32,4158,49461,6774,702,458,7373,9315,11,400,51,15159,68,51185,20,15,11,15,15,15,3,12(...TRUNCATED) | 4.5e33 | "<|im_start|>user\nA white dwarf star has an effective temperature, $T_{e}=50,000$ degrees Kelvin, b(...TRUNCATED) | [[32313,11,773,358,1184,311,12564,279,44958,22053,315,264,4158,49461,6774,13,576,3491,6696,752,279,7(...TRUNCATED) | sharding_1_2025-07-16 23:36.jsonl | minerva_math | 18,598 |
["Okay, so I need to find the critical angle for the light to be internally reflected at the hypoten(...TRUNCATED) | [151644,872,198,47,86706,25,362,93610,374,20346,504,8991,323,702,11067,429,1352,264,1290,21495,448,2(...TRUNCATED) | 41.8 | "<|im_start|>user\nPreamble: A prism is constructed from glass and has sides that form a right trian(...TRUNCATED) | [[32313,11,773,358,1184,311,1477,279,9023,9210,369,279,3100,311,387,33067,25911,518,279,9751,65628,8(...TRUNCATED) | sharding_1_2025-07-16 23:36.jsonl | minerva_math | 7,404 |
["Okay, so I need to find the apparent magnitude of a star that has an absolute magnitude of -7 when(...TRUNCATED) | [151644,872,198,32,3953,6774,702,458,10740,25621,400,44,10829,22,12947,1416,419,6774,374,13166,304,2(...TRUNCATED) | 20.39 | "<|im_start|>user\nA particular star has an absolute magnitude $M=-7$. If this star is observed in a(...TRUNCATED) | [[32313,11,773,358,1184,311,1477,279,9981,25621,315,264,6774,429,702,458,10740,25621,315,481,22,979,(...TRUNCATED) | sharding_1_2025-07-16 23:36.jsonl | minerva_math | 12,104 |
["Okay, so I need to find the gravitational acceleration due to the Sun at the Earth's orbit, which (...TRUNCATED) | [151644,872,198,9885,279,69919,30803,4152,311,279,8059,518,279,3728,315,279,9237,594,26507,320,72,17(...TRUNCATED) | 0.006 | "<|im_start|>user\nFind the gravitational acceleration due to the Sun at the location of the Earth's(...TRUNCATED) | [[32313,11,773,358,1184,311,1477,279,69919,30803,4152,311,279,8059,518,279,9237,594,26507,11,892,374(...TRUNCATED) | sharding_1_2025-07-16 23:36.jsonl | minerva_math | 14,566 |
["Okay, so I need to find the critical angle for the water-air interface. Hmm, the critical angle is(...TRUNCATED) | [151644,872,198,47,86706,25,362,4530,7145,3100,23418,16992,1095,304,3015,374,10455,389,279,7329,320,(...TRUNCATED) | np.arcsin(10/13) | "<|im_start|>user\nPreamble: A collimated light beam propagating in water is incident on the surface(...TRUNCATED) | [[32313,11,773,358,1184,311,1477,279,9023,9210,369,279,3015,37535,3749,13,88190,11,279,9023,9210,374(...TRUNCATED) | sharding_1_2025-07-16 23:36.jsonl | minerva_math | 1,784 |
["Okay, so I need to find the theoretical limiting angular resolution of an 8-inch diameter optical (...TRUNCATED) | [151644,872,198,9885,279,31787,32894,20314,10935,320,258,15580,5024,8,315,264,8353,220,23,24241,320,(...TRUNCATED) | 0.49 | "<|im_start|>user\nFind the theoretical limiting angular resolution (in arcsec) of a commercial 8-in(...TRUNCATED) | [[32313,11,773,358,1184,311,1477,279,31787,32894,20314,10935,315,458,220,23,24241,23033,28293,55825,(...TRUNCATED) | sharding_1_2025-07-16 23:36.jsonl | minerva_math | 17,089 |
["Okay, so I need to figure out how far away a star is if its parallax is 0.01 arcseconds. Hmm, I re(...TRUNCATED) | [151644,872,198,32,6774,702,264,16878,1346,44168,315,400,15,13,15,16,61,35702,32338,1124,32338,31716(...TRUNCATED) | 100 | "<|im_start|>user\nA star has a measured parallax of $0.01^{\\prime \\prime}$, that is, $0.01$ arcse(...TRUNCATED) | [[32313,11,773,358,1184,311,7071,700,1246,3041,3123,264,6774,374,421,1181,1346,44168,374,220,15,13,1(...TRUNCATED) | sharding_1_2025-07-16 23:36.jsonl | minerva_math | 1,762 |
["Okay, so I need to figure out the decrease in magnitude of a star-planet system when the planet tr(...TRUNCATED) | [151644,872,198,2082,36275,7417,11580,702,1012,13166,892,16211,304,4065,315,320,72,1734,2572,1356,11(...TRUNCATED) | 0.022 | "<|im_start|>user\nAn extrasolar planet has been observed which passes in front of (i.e., transits) (...TRUNCATED) | [[32313,11,773,358,1184,311,7071,700,279,18472,304,25621,315,264,6774,12,50074,1849,979,279,11580,13(...TRUNCATED) | sharding_1_2025-07-16 23:36.jsonl | minerva_math | 13,952 |
["Okay, so I need to figure out the wavelength of a photon emitted when an electron transitions from(...TRUNCATED) | [151644,872,198,2679,279,51350,81,4802,5866,5452,438,400,57,47822,17,31716,11,1380,400,57,3,374,279,(...TRUNCATED) | 9.6 | "<|im_start|>user\nIf the Bohr energy levels scale as $Z^{2}$, where $Z$ is the atomic number of the(...TRUNCATED) | [[32313,11,773,358,1184,311,7071,700,279,45306,315,264,67910,46942,979,458,16715,33592,504,308,28,18(...TRUNCATED) | sharding_1_2025-07-16 23:36.jsonl | minerva_math | 18,306 |
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