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Solving Scale Problems

Solving Scale Problems. The solution will gradually “build” for each problem. Some unit conversions. 1 m = 100 cm = 1000 mm 1 km = 1000 m 1 foot = 12 inches (1’ = 12”) 1 mile = 63,360 inches 1 hectare (ha) = 10,000m 2.

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Solving Scale Problems

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  1. Solving Scale Problems The solution will gradually “build” for each problem

  2. Some unit conversions • 1 m = 100 cm = 1000 mm • 1 km = 1000 m • 1 foot = 12 inches (1’ = 12”) • 1 mile = 63,360 inches • 1 hectare (ha) = 10,000m2

  3. The real world distance between 2 features is 1600m and the map distance is 8 cm.What is the map scale ? • Scale = 1 / xxxxx = map dist / real world distance • If we know 3 of the figures, then we solve for the unknown by cross-multiply & then divide • What are we solving for? • Map scale is unknown • 1 / xxx = map dist. / real world (RW) dist • What do we know?: • Map dist = 8 cm • RW distance is 1,600 m = 160,000 cm • 1 / xxx = 8 cm / 160,000 cm • xxx = (1*160,000 cm)/8 cm • xxx = 20,000 • Scale is 1 : 20,000

  4. The distance between 2 features on a 1:10,000 map is 5 cm. What is the real world distance? • Scale = 1 / xxxx = map dist / real world distance • If we know 3 of the figures, then we solve for the unknown by cross-multiply & then divide • What are we solving for? • Real world (RW) distance • 1 /xxxx = map dist / RW dist • What do we know?: • Scale of map (1/xxxx) is 1 / 10,000 • Map dist = 5 cm • 1 / 10,000 = 5 cm / RW dist. cm • RW dist (cm) = (5*10,000 cm)/1 • RW dist cm = 50,000 cm • In metres, RW dist = 500 m

  5. What is the length in cm of one km on a 1:25,000 map? • Scale = 1 / xxxxx = map dist / real world distance • If we know 3 of the figures, then we solve for the unknown by cross-multiply & then divide • What are we solving for? • Length on map (in cm) • 1 / x = map dist. / real world (RW) dist • What do we know?: • Scale of map (1/x) is 1 / 25,000 • 1 km RW dist (1 km = 1,000 m = 100,000 cm) • 1 / 25,000 = map dist cm / 100,000 cm • Map dist (cm) = (1*100,000 cm)/25,000 • Map dist = 4 cm

  6. You have a photo with a scale expressed as 1” = ¼ mile. What is the scale as a representative fraction? • This is really a unit conversion question • Scale = 1 / xxx = map / RW • Scale of photo is 1” = ¼ mile • If you convert top and bottom values to the same units, then you have a representative fraction (since the units will cancel out) • So the question really is … how many inches in ¼ mile? • 1 mile = 63,360 inches • ¼ mile = 63,360 inches / 4 = 15,840 inches • Scale is 1 : 15,840 (NO need to show units anymore)

  7. A map scale is 1 cm = 750 m and a photo covers a portion of the map area.  The distance between two road intersections measures 9.8 cm on the photo and 5.0 cm on the map.  Determine the scale of the photo. • Photo scale = 1/xxx = photo/RW • We know • photo measure of 9.8 cm • If we had the RW distance we could determine the photo scale. • Can we find RW distance? • Yes, as we have map scale and a map measure • So first we find RW distance from map, then use it to solve for photo scale • Map scale is • 1 cm = 750 m, or • 1 cm = 75,000 cm, or • 1 : 75,000 • RW distance is • RW = 5.0 cm * 75,000 • RW = 375,000 cm

  8. A map scale is 1 cm = 750 m and a photo covers a portion of the map area.  The distance between two road intersections measures 9.8 cm on the photo and 5.0 cm on the map.  Determine the scale of the photo. • From last slide • Photo scale is … • RW distance (from map measures) is • RW = 5.0 cm * 75,000 • RW = 375,000 cm • Realize that RW distance is the same no matter if we calculated from a map or a photo – the RW distance is the same • Now we can solve for photo scale (xxxx) • xxxx = (1 * 375,000cm) / 9.8 cm • xxxx = 38,265 • Scale is 1 : 38,265

  9. Solve for Area • The unit we will use for an area measure is the hectare (ha) • 1 ha = 10,000 m2 • You can visualize this as a large square that measures 100m on each side.

  10. How many hectares are covered by a standard air photo at a scale of 1:20,000 • First remember that an air photo measures 9” by 9” • We want to know what ground area is covered in the photo. • First, since we will be solving for a metric area measure (ha), it is best to first convert 9” to cm • Photo edge = 9 in • = 9 in * 2.54 cm/in • = 22.86 cm • Second, we find the RW distance in metres • RW = (20,000*22.86cm) / 1 • RW = 457200 cm = 4572 m • Third, we find area (L * W) • Area = 4572m * 4572m • Area = 20903184m2 • Area = 2090 ha

  11. How many hectares are covered by a standard air photo at a scale of 1” = ¼ mile • The photo scale 1” = ¼ mi. needs to be converted to a representative fraction • Solve by converting ¼ mi to inches • 63,360 in/mi / 4 = 15,840” • Next, convert the photo edge (9”) to metric (cm) • Since we will be solving for a metric measure (ha), it is best to first convert 9” to cm • Photo edge = 9 in • = 9 in * 2.54 cm/in • = 22.86 cm • Now we find the RW distance of the photo edge • RW cm = (22.86 cm*15840)/1 • RW = 362,102 cm = 3621 m • Last we calculate area, in m2, then convert to ha • Area = 3621 m * 3621 m = 13,111,641 m2 • Area = 13,111,641 m2 / 10,000 = 1,311 ha

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