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

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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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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 1=Map dist. xxxxReal world dist. 1=Map dist. xxxxReal world dist. 1= 8cm xxxx160,000cm

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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 1=Map dist. xxxxReal world dist. 1=Map dist. xxxxReal world dist. 1=5 cm 10,000Real world dist. cm

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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 1=Map dist. xxxxReal world dist. 1=Map dist. xxxxReal world dist. 1=Map dist. cm 25,000100,000cm

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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) 1=Map dist. xxxxReal world dist. 1=1 inch xxxx¼ mile 1=1 in xxxx15,840 in 1=1 xxxx15,840

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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 1=9.8 cm xxxxRW 1=5.0 cm 75,000RW 1=Photo xxxxRW

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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 1=9.8 cm xxxxRW cm 1=9.8 cm xxxx375000 cm

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Solve for Area The unit we will use for an area measure is the hectare (ha) 1 ha = 10,000 m 2 You can visualize this as a large square that measures 100m on each side.

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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 = 20903184m 2 –Area = 2090 ha 1=22.86 cm 20000RW cm

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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 m 2, then convert to ha –Area = 3621 m * 3621 m = 13,111,641 m 2 –Area = 13,111,641 m 2 / 10,000 = 1,311 ha 1=1” xxxx¼ mi 1=1” xxxx15,840” 1=22.86 cm 15840RW cm

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