Summary of The SAT Question Everyone Got Wrong

• 00:00:00

  In 1982, there was a SAT question that every student got wrong. The question was about the number of revolutions a smaller circle would make while rolling around a larger circle. The test writers themselves got the question wrong, and the correct answer was not among the provided options on the test. Only three students wrote to the College Board to point out the error, and they were proven correct. The SAT has a reputation for determining students' futures, and mistakes like this were not supposed to happen. The three students argued that the wording of the question was unclear and led to the incorrect answers.

• 00:03:14

  In this video, the narrator explains a paradox involving two identical coins rolling around each other. Initially, it is expected that the coin would rotate once, but it actually rotates twice. This paradox also applies to a similar question in the SAT exam, where the correct answer is four rotations. The explanation involves considering the ratio of the circumferences of the two circles and adding one extra rotation for the circular path traveled. However, there are other perspectives to consider. From the perspective of circle A, it rotates three times to return to its starting position, while from the perspective of the coins, the outer coin only rotates once. The video also mentions that the definition of a revolution in astronomy is precise.

• 00:07:08

  Circle A only revolves around circle B once, as it completes a full orbit around another body. There are multiple definitions of a revolution, some including an object rotating about its own axis. The question in the video was ambiguous, leading to multiple possible answers. The College Board acknowledged their mistake and nullified the question for all test takers. The explanation involves understanding that the amount the small circle rotates is always equal to the distance the center travels. This can be proven using the concept of relative velocities. In this specific problem, the small circle rotates four times, matching the total distance the center moves around a circle with a radius of four.

• 00:10:03

  When a circular object rolls without slipping, the distance its center travels is equal to the amount it turns. This applies to circles rolling on any surface, whether it be on the outside or inside of a shape. To determine the number of rotations made, divide the distance traveled by the circle's circumference. This concept is more general than the answer to the coin paradox, where adding one to the expected answer reveals the shortcut. If a circle is rolling continuously around a shape, the distance traveled by the center is the perimeter of the shape plus the circle's circumference. Dividing this by the circle circumference gives N plus one rotations. If a circle is rolling continuously within a shape, the distance traveled by the center decreases by one circumference of the circle, resulting in N minus one rotations. If the circle is rolling along a flat line, the distance traveled by the center equals the length of the line, which is N rotations. This principle extends beyond mathematics and is essential in astronomy. When counting the number of days in a year, we count how many rotations the Earth makes in its orbit around the sun. However, from an external observer's perspective, the Earth does one extra rotation to account for its circular path around the sun. This discrepancy is known as the Sidereal year. A normal solar day is the time it takes for the sun to be directly overhead again on Earth. Since Earth is also orbiting the sun, it must rotate more than 360 degrees in a 24-hour solar day. However, when it comes to distant stars, Earth only needs to rotate exactly 360 degrees to see a star directly overhead again. Therefore, a Sidereal day is slightly shorter, lasting 23 hours, 56 minutes, and four seconds.

• 00:13:08

  In summary, this video explains the concept of the extra day in the Sidereal year, where the Sidereal day gradually diverges from the solar day throughout the year. It discusses the use of Sidereal time in astronomy for tracking objects in space and for geostationary satellites. The video also touches on the 1982 SAT question that was rescinded, affecting students' scores and potentially their educational opportunities. It mentions the decline of standardized testing in college admissions and the narrator's personal experience with math and writing math questions.

• 00:16:19

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