Scale, distance and exact location
Grade 9 Social Studies Worksheet - Challenge Me
Circle the best answer.
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Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km.
Why does the great-circle route from Phoenix to Charleston bend on a Mercator map, Ari asks?
- The route really bends, since Earth’s surface bends the shortest path
- Mercator keeps constant headings straight, so the shortest path curves
- A Mercator map keeps the shortest paths straight and bends compass headings
- A Mercator map is flat, so any route drawn on it must bend
A captain holds one compass heading between two ports. How does that route compare with the great-circle route, Bo asks?
- It is the same length, since a straight line on the map is the shortest
- It is shorter, since a constant heading avoids the bend in the great circle
- It is never shorter, since the great circle is the shortest path on a sphere
- It is longer in every case, so the two never match
Two ports lie on 40°N. A ship sails due east along the 40°N parallel. Is that the shortest route, Esme asks?
- No: the shortest route bulges south of the line toward the equator
- No: the shortest route bulges north of the 40°N line toward the pole
- Yes: each parallel is a great circle, so it is the shortest route
- Yes: staying on one latitude holds one heading, which is the shortest
Which comparison of Phoenix–Quito (5,130 km) and Phoenix–Charleston (2,780 km) is fair, Jun asks?
- About 1.8 times; one sphere gives both, so the ratio is steadier than either
- No ratio can be trusted, since Earth is not a perfect sphere and both distances are off
- Quito is twice as far, since 5,130 is about double 2,780, a ratio of two to one
- Quito is exactly 1.85 times as far, since the two figures are exact to the kilometer
Use this
Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km. Card B. A flat-plane check. Using the city coordinates, a student gets Phoenix to Charleston as 2,794 km with a right triangle on a flat map. Coordinates are rounded to the minute, and one minute of latitude is about 1.85 km.
A flat-plane method gave 2,794 km, and the sphere gave 2,780 km. What does the 14 km gap show, Kira asks?
- The sphere figure is wrong by 14 km, so the flat figure is right
- The gap shows Earth is flat over distances of 2,800 km
- The two must agree exactly, so one of them has an error
- A flat triangle is close here: 14 km is about half a percent
A report gives Phoenix to Charleston as 2,780.4 km. What is wrong with the extra decimal, Luz asks?
- The decimal should be kilometers, 2,780 km and 400 m
- The distance cannot be given, since rounding spoils the whole result
- Each end could shift about 1 km, so “about 2,780 km” is honest
- Nothing: 2,780.4 km is just 2,780 km with the decimal for accuracy
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Card C. East–west distance. One degree of longitude spans about 111 km times the cosine of the latitude. The cosine of 60° is 0.50. Two towns lie on the 60°N parallel, 10° of longitude apart.
Two towns on the 60°N parallel are 10° of longitude apart. How far apart are they along the parallel, Milo asks?
- About 55 km: 10° at 5.5 km a degree
- About 1,110 km: 10° at 111 km a degree
- About 555 km: 10° at 111 × 0.50 km a degree
- About 1,110 km along the parallel, and 555 km in a straight line
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Card D. Drawing roads. On a 1:1,000,000 map, 1 mm stands for 1 km. A country road about 10 m wide is drawn as a line at least 0.3 mm wide.
Why is a 10 m road drawn 0.3 mm wide on a 1:1,000,000 map, Pia asks?
- At true scale it would be 0.01 mm, too thin to see, so it is widened
- Widening a line makes the map’s scale wrong everywhere on it
- At true scale it would be 1 mm, so drawing it 0.3 mm shrinks it
- The road is really 300 m wide, which is what 0.3 mm stands for
Use this
Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km.
Tala’s note says: “On a Mercator map the shortest route is always a straight line.” Which fix is right?
- Change it: the shortest route is straight on a globe, so it is straight on a map
- Leave it: Mercator is accurate, so a straight line is the shortest
- Change it: the shortest route is straight, and a constant heading curves
- Change it: a constant compass heading is straight; the shortest route usually curves
Use this
Card A. Routes, in our words, for a round Earth of radius 6,371 km. A great-circle route is the shortest path between two points on a sphere. A route that keeps one compass heading is a straight line on a Mercator map. Between two places on the same latitude north of the equator, the great-circle route bulges toward the pole. Computed great-circle distances: Phoenix to Charleston, about 2,780 km; Phoenix to Quito, about 5,130 km. Card B. A flat-plane check. Using the city coordinates, a student gets Phoenix to Charleston as 2,794 km with a right triangle on a flat map. Coordinates are rounded to the minute, and one minute of latitude is about 1.85 km.
Zee’s note says: “Phoenix to Quito is 5,130 km, so Quito is 5,130 km south of Phoenix.” Which fix is right?
- Leave it: Quito is directly south, so the two distances match
- Change it: 5,130 km is direct; the south part is about 3,740 km
- Change it: Quito is 5,130 km east of Phoenix, not south
- Change it: the south part is 5,130 + 544 km, adding Charleston’s gap