Scale, distance and exact location

Grade 9 Social Studies Worksheet - Challenge Me

Circle the best answer.

1. Read this first

   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.

2. 1.

   A captain holds one compass heading between two ports. How does that route compare with the great-circle route, Bo asks?

   • It is never shorter, since the great circle is the shortest path on a sphere
   • It is shorter, since a constant heading avoids the bend in the great circle
   • It is the same length, since a straight line on the map is the shortest
   • It is longer in every case, so the two never match
3. 2.

   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
   • Yes: each parallel is a great circle, so it is the shortest route
   • No: the shortest route bulges north of the 40°N line toward the pole
   • Yes: staying on one latitude holds one heading, which is the shortest

4. 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.

5. 3.

   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 gap shows Earth is flat over distances of 2,800 km
   • The sphere figure is wrong by 14 km, so the flat figure is right
   • A flat triangle is close here: 14 km is about half a percent
   • The two must agree exactly, so one of them has an error
6. 4.

   A report gives Phoenix to Charleston as 2,780.4 km. What is wrong with the extra decimal, Luz asks?

   • Each end could shift about 1 km, so “about 2,780 km” is honest
   • The distance cannot be given, since rounding spoils the whole result
   • Nothing: 2,780.4 km is just 2,780 km with the decimal for accuracy
   • The decimal should be kilometers, 2,780 km and 400 m

7. Use this

   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.

8. 5.

   Two towns on the 60°N parallel are 10° of longitude apart. How far apart are they along the parallel, Milo asks?

   • 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
   • About 55 km: 10° at 5.5 km a degree
   • About 1,110 km: 10° at 111 km a degree

9. Use this

   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. 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.

10. 6.

    For those two towns, how does the great-circle distance compare with the 555 km along the parallel, Noah asks?

    • It is half as long, since the pole shortens the path
    • It is the same, since both stay on the 60°N line
    • A little shorter: the great circle bulges toward the pole
    • It is longer, since the great circle bulges and adds distance

11. Use this

    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.

12. 7.

    Why is a 10 m road drawn 0.3 mm wide on a 1:1,000,000 map, Pia asks?

    • Widening a line makes the map’s scale wrong everywhere on it
    • The road is really 300 m wide, which is what 0.3 mm stands for
    • At true scale it would be 0.01 mm, too thin to see, so it is widened
    • At true scale it would be 1 mm, so drawing it 0.3 mm shrinks it
13. 8.

    Which measurement does drawing roads at 0.3 mm distort most on that map, Ren asks?

    • The map’s ratio scale, since line widths are part of a ratio
    • A road’s width; the distance between two town centers stays reliable
    • The distance between two town centers, since a wide line moves the centers
    • The length of the road, since a thick line is longer than a thin one

14. 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.

15. 9.

    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
    • Change it: a constant compass heading is straight; the shortest route usually curves
    • Leave it: Mercator is accurate, so a straight line is the shortest
    • Change it: the shortest route is straight, and a constant heading curves

16. 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.

17. 10.

    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: the south part is 5,130 + 544 km, adding Charleston’s gap
    • 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