id int64 1 14M | domain stringclasses 6
values | topic stringclasses 23
values | subtopic stringclasses 37
values | difficulty int64 1 8 | unit_type stringclasses 3
values | title stringlengths 14 86 | content stringlengths 203 553 | key_equations stringclasses 23
values | prerequisites stringclasses 29
values | learning_objective stringclasses 37
values |
|---|---|---|---|---|---|---|---|---|---|---|
1,401 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 10.9 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 10.9 AU one obtains T = 36.01 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,402 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 11.55 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 11.55 AU one obtains T = 39.24 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,403 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 9.807 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 9.807 AU one obtains T = 30.71 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,404 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 2.366 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 2.366 AU one obtains T = 3.638 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,405 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 23.04 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 23.04 AU one obtains T = 110.6 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,406 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 33.66 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 33.66 AU one obtains T = 195.3 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,407 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 6.482 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 6.482 AU one obtains T = 16.5 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,408 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 14.69 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 14.69 AU one obtains T = 56.3 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,409 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 17.33 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 17.33 AU one obtains T = 72.16 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,410 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 12.06 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 12.06 AU one obtains T = 41.9 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,411 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 26.62 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 26.62 AU one obtains T = 137.3 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,412 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 24.17 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 24.17 AU one obtains T = 118.8 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,413 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 8.307 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 8.307 AU one obtains T = 23.94 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,414 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 1.419 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 1.419 AU one obtains T = 1.691 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,415 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 7.165 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 7.165 AU one obtains T = 19.18 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,416 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 11.96 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 11.96 AU one obtains T = 41.34 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,417 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 3.645 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 3.645 AU one obtains T = 6.958 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,418 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 33.81 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 33.81 AU one obtains T = 196.6 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,419 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 12.61 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 12.61 AU one obtains T = 44.78 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,420 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 16.14 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 16.14 AU one obtains T = 64.84 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,421 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 19.77 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 19.77 AU one obtains T = 87.89 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,422 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 26.58 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 26.58 AU one obtains T = 137 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,423 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 4.009 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 4.009 AU one obtains T = 8.027 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,424 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 21.95 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 21.95 AU one obtains T = 102.8 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,425 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 7.721 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 7.721 AU one obtains T = 21.46 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,426 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 35.47 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 35.47 AU one obtains T = 211.2 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,427 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 15.03 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 15.03 AU one obtains T = 58.26 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,428 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 18.05 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 18.05 AU one obtains T = 76.7 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,429 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 10.83 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 10.83 AU one obtains T = 35.62 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,430 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 18.82 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 18.82 AU one obtains T = 81.62 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,431 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 9.359 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 9.359 AU one obtains T = 28.63 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,432 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 11.04 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 11.04 AU one obtains T = 36.66 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,433 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 2.841 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 2.841 AU one obtains T = 4.788 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,434 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 30.19 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 30.19 AU one obtains T = 165.9 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,435 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 26.83 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 26.83 AU one obtains T = 139 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,436 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 3.794 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 3.794 AU one obtains T = 7.39 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,437 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 14.01 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 14.01 AU one obtains T = 52.46 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,438 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 21.84 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 21.84 AU one obtains T = 102.1 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,439 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 38.84 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 38.84 AU one obtains T = 242.1 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,440 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 23.75 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 23.75 AU one obtains T = 115.8 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,441 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 22.32 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 22.32 AU one obtains T = 105.5 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,442 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 33.7 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 33.7 AU one obtains T = 195.6 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,443 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 32.81 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 32.81 AU one obtains T = 187.9 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,444 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 16.98 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 16.98 AU one obtains T = 69.96 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,445 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 21.61 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 21.61 AU one obtains T = 100.4 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,446 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 34.73 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 34.73 AU one obtains T = 204.7 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,447 | earth_space | astronomy | kepler_third_law | 5 | worked_example | Orbital period for semi-major axis 19.2 AU | For a planet orbiting the Sun, Kepler's third law states that the square of the sidereal orbital period T (in years) equals the cube of the semi-major axis a (in AU): T² = a³. With a = 19.2 AU one obtains T = 84.15 years. The law is a direct consequence of Newtonian gravity for a central inverse-square force. | T^2 = a^3 (solar units) | newtonian gravity | Apply Kepler's third law to relate orbital period and semi-major axis. |
1,448 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 3.817 x + -0.9495 at x = -8.421 | The linear relation y = m x + b with slope m = 3.817 and intercept b = -0.9495 evaluated at x = -8.421 yields y = -33.09. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,449 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 4.028 x + 8.572 at x = 0.04203 | The linear relation y = m x + b with slope m = 4.028 and intercept b = 8.572 evaluated at x = 0.04203 yields y = 8.742. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,450 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 4.005 x + 12.02 at x = 3.551 | The linear relation y = m x + b with slope m = 4.005 and intercept b = 12.02 evaluated at x = 3.551 yields y = 26.24. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,451 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 1.203 x + -15.19 at x = 5.144 | The linear relation y = m x + b with slope m = 1.203 and intercept b = -15.19 evaluated at x = 5.144 yields y = -9.002. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,452 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -3.271 x + 19.37 at x = 9.443 | The linear relation y = m x + b with slope m = -3.271 and intercept b = 19.37 evaluated at x = 9.443 yields y = -11.52. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,453 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 3.085 x + -14.95 at x = -1.533 | The linear relation y = m x + b with slope m = 3.085 and intercept b = -14.95 evaluated at x = -1.533 yields y = -19.68. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,454 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 4.883 x + -2.584 at x = 9.945 | The linear relation y = m x + b with slope m = 4.883 and intercept b = -2.584 evaluated at x = 9.945 yields y = 45.98. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,455 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 1.272 x + 13.36 at x = -4.842 | The linear relation y = m x + b with slope m = 1.272 and intercept b = 13.36 evaluated at x = -4.842 yields y = 7.204. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,456 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 4.108 x + 16.57 at x = -8.66 | The linear relation y = m x + b with slope m = 4.108 and intercept b = 16.57 evaluated at x = -8.66 yields y = -19.01. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,457 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -1.119 x + -4.106 at x = -3.477 | The linear relation y = m x + b with slope m = -1.119 and intercept b = -4.106 evaluated at x = -3.477 yields y = -0.2145. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,458 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -2.238 x + -1.674 at x = 7.469 | The linear relation y = m x + b with slope m = -2.238 and intercept b = -1.674 evaluated at x = 7.469 yields y = -18.39. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,459 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 2.866 x + 4.922 at x = 0.4512 | The linear relation y = m x + b with slope m = 2.866 and intercept b = 4.922 evaluated at x = 0.4512 yields y = 6.215. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,460 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -0.8041 x + -3.422 at x = -7.036 | The linear relation y = m x + b with slope m = -0.8041 and intercept b = -3.422 evaluated at x = -7.036 yields y = 2.235. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,461 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 0.8793 x + 10.34 at x = 8.793 | The linear relation y = m x + b with slope m = 0.8793 and intercept b = 10.34 evaluated at x = 8.793 yields y = 18.07. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,462 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 4.249 x + 2.507 at x = -7.98 | The linear relation y = m x + b with slope m = 4.249 and intercept b = 2.507 evaluated at x = -7.98 yields y = -31.4. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,463 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -2.139 x + 1.425 at x = -3.123 | The linear relation y = m x + b with slope m = -2.139 and intercept b = 1.425 evaluated at x = -3.123 yields y = 8.105. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,464 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -0.8911 x + -4.678 at x = -0.2883 | The linear relation y = m x + b with slope m = -0.8911 and intercept b = -4.678 evaluated at x = -0.2883 yields y = -4.421. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,465 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 1.09 x + -18.5 at x = -4.492 | The linear relation y = m x + b with slope m = 1.09 and intercept b = -18.5 evaluated at x = -4.492 yields y = -23.4. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,466 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -3.561 x + 4.346 at x = 3.873 | The linear relation y = m x + b with slope m = -3.561 and intercept b = 4.346 evaluated at x = 3.873 yields y = -9.448. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,467 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -4.612 x + 15.58 at x = -3.37 | The linear relation y = m x + b with slope m = -4.612 and intercept b = 15.58 evaluated at x = -3.37 yields y = 31.13. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,468 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -2.624 x + 9.83 at x = 8.417 | The linear relation y = m x + b with slope m = -2.624 and intercept b = 9.83 evaluated at x = 8.417 yields y = -12.26. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,469 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 3.97 x + -19.19 at x = 6.348 | The linear relation y = m x + b with slope m = 3.97 and intercept b = -19.19 evaluated at x = 6.348 yields y = 6.012. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,470 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -1.969 x + -8.786 at x = -0.1676 | The linear relation y = m x + b with slope m = -1.969 and intercept b = -8.786 evaluated at x = -0.1676 yields y = -8.456. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,471 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 1.962 x + -16.07 at x = 7.378 | The linear relation y = m x + b with slope m = 1.962 and intercept b = -16.07 evaluated at x = 7.378 yields y = -1.597. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,472 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -3.656 x + 18.97 at x = -1.138 | The linear relation y = m x + b with slope m = -3.656 and intercept b = 18.97 evaluated at x = -1.138 yields y = 23.13. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,473 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 3.258 x + -9.223 at x = -1.665 | The linear relation y = m x + b with slope m = 3.258 and intercept b = -9.223 evaluated at x = -1.665 yields y = -14.65. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,474 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 1.456 x + -12.48 at x = -5.772 | The linear relation y = m x + b with slope m = 1.456 and intercept b = -12.48 evaluated at x = -5.772 yields y = -20.88. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,475 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 3.24 x + 9.638 at x = 5.19 | The linear relation y = m x + b with slope m = 3.24 and intercept b = 9.638 evaluated at x = 5.19 yields y = 26.45. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,476 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 3.672 x + 12.84 at x = 0.3054 | The linear relation y = m x + b with slope m = 3.672 and intercept b = 12.84 evaluated at x = 0.3054 yields y = 13.96. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,477 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -3.41 x + -7.555 at x = 0.1359 | The linear relation y = m x + b with slope m = -3.41 and intercept b = -7.555 evaluated at x = 0.1359 yields y = -8.019. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,478 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -3.644 x + 14.05 at x = 7.587 | The linear relation y = m x + b with slope m = -3.644 and intercept b = 14.05 evaluated at x = 7.587 yields y = -13.59. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,479 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -4.711 x + -12.29 at x = 6.659 | The linear relation y = m x + b with slope m = -4.711 and intercept b = -12.29 evaluated at x = 6.659 yields y = -43.65. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,480 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 3.37 x + -10.02 at x = -0.871 | The linear relation y = m x + b with slope m = 3.37 and intercept b = -10.02 evaluated at x = -0.871 yields y = -12.96. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,481 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 4.18 x + 8.185 at x = -4.52 | The linear relation y = m x + b with slope m = 4.18 and intercept b = 8.185 evaluated at x = -4.52 yields y = -10.71. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,482 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 3.235 x + 0.2051 at x = 2.708 | The linear relation y = m x + b with slope m = 3.235 and intercept b = 0.2051 evaluated at x = 2.708 yields y = 8.966. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,483 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -3.761 x + -18.78 at x = -2.551 | The linear relation y = m x + b with slope m = -3.761 and intercept b = -18.78 evaluated at x = -2.551 yields y = -9.184. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,484 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 0.9404 x + -12.9 at x = 7.41 | The linear relation y = m x + b with slope m = 0.9404 and intercept b = -12.9 evaluated at x = 7.41 yields y = -5.928. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,485 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 0.8688 x + -6.01 at x = -6.73 | The linear relation y = m x + b with slope m = 0.8688 and intercept b = -6.01 evaluated at x = -6.73 yields y = -11.86. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,486 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 3.945 x + 9.958 at x = 3.777 | The linear relation y = m x + b with slope m = 3.945 and intercept b = 9.958 evaluated at x = 3.777 yields y = 24.86. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,487 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -2.149 x + -4.537 at x = -6.742 | The linear relation y = m x + b with slope m = -2.149 and intercept b = -4.537 evaluated at x = -6.742 yields y = 9.953. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,488 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 0.7226 x + 18.6 at x = 7.142 | The linear relation y = m x + b with slope m = 0.7226 and intercept b = 18.6 evaluated at x = 7.142 yields y = 23.76. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,489 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 1.474 x + 7.108 at x = -4.618 | The linear relation y = m x + b with slope m = 1.474 and intercept b = 7.108 evaluated at x = -4.618 yields y = 0.3013. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,490 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -0.9049 x + -19.2 at x = 5.606 | The linear relation y = m x + b with slope m = -0.9049 and intercept b = -19.2 evaluated at x = 5.606 yields y = -24.27. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,491 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 2.676 x + -19.64 at x = 8.23 | The linear relation y = m x + b with slope m = 2.676 and intercept b = -19.64 evaluated at x = 8.23 yields y = 2.378. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,492 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 1.474 x + 4.046 at x = -9.831 | The linear relation y = m x + b with slope m = 1.474 and intercept b = 4.046 evaluated at x = -9.831 yields y = -10.44. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,493 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -2.476 x + 12.2 at x = -3.891 | The linear relation y = m x + b with slope m = -2.476 and intercept b = 12.2 evaluated at x = -3.891 yields y = 21.84. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,494 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 4.67 x + 5.71 at x = -1.524 | The linear relation y = m x + b with slope m = 4.67 and intercept b = 5.71 evaluated at x = -1.524 yields y = -1.407. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,495 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -1.235 x + -6.052 at x = -4.96 | The linear relation y = m x + b with slope m = -1.235 and intercept b = -6.052 evaluated at x = -4.96 yields y = 0.07522. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,496 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -0.333 x + 7.088 at x = 6.486 | The linear relation y = m x + b with slope m = -0.333 and intercept b = 7.088 evaluated at x = 6.486 yields y = 4.928. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,497 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -1.028 x + -15.91 at x = 0.2303 | The linear relation y = m x + b with slope m = -1.028 and intercept b = -15.91 evaluated at x = 0.2303 yields y = -16.14. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,498 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 1.624 x + 13.73 at x = -2.515 | The linear relation y = m x + b with slope m = 1.624 and intercept b = 13.73 evaluated at x = -2.515 yields y = 9.647. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,499 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = 1.473 x + 4.362 at x = -4.03 | The linear relation y = m x + b with slope m = 1.473 and intercept b = 4.362 evaluated at x = -4.03 yields y = -1.577. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
1,500 | mathematics | algebra | linear_relation | 2 | worked_example | Evaluate linear function y = -3.919 x + -17.45 at x = 9.767 | The linear relation y = m x + b with slope m = -3.919 and intercept b = -17.45 evaluated at x = 9.767 yields y = -55.72. Linear models appear throughout science whenever a rate of change is approximately constant. | y = m x + b | basic arithmetic | Evaluate and interpret a linear function in a scientific context. |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.