41. Trapezoid. A four-sided figure with two parallel sides is called a trapezoid. 42. Area of a trapezoid. If a trapezoid T have a duplicate + T Н D cut from paper and turned over and fitted to it, like D, the two together make a parallelogram. (Illustrate by paper cutting.) Therefore The area of a trapezoid equals half that of a rectangle with the same altitude and with a base equal to the sum of the two parallel sides. 43. Illustrative problem. What is the area of a trapezoid of altitude 4 in. and parallel sides 8 in. and 10 in.? As above explained, the area is of 4 times (8 + 10) sq. in., or 36 sq. in. ORAL EXERCISE State the areas of the trapezoids whose altitudes are first given in Exs. 1-10, followed by the two parallel sides: 10. 11 in., 114 in. and 183 in. 11. If the area of a trapezoid is 48 sq. in., and the parallel sides are 5 in. and 7 in., what is the altitude? 12. If the area of a trapezoid is 48 sq. in. and the altitude is 6 in., what is the sum of the parallel sides? If one of these sides is 9 in., what is the other? WRITTEN EXERCISE Find the areas of the trapezoids whose altitudes are first given in Exs. 1-7, followed by the two parallel sides: 1. 5 rd., 6 rd. 7 ft. and 9 rd. 2. 322 ft., 427 ft. and 534 ft. 3. 127 ft., 129 ft. and 148 ft. 4. 236 in., 208 in. and 235 in. 5. 34 ft., 27 ft. 8 in. and 33 ft. 6. 62 ft. 3 in., 59 ft. and 78 ft. 7. 4 yd., 2 yd. 27 in. and 6 yd. 8. If the area of a trapezoid is 21 sq. ft. 16 sq. in., the altitude 5 ft. 4 in., and one of the parallel sides 4 ft. 9 in., how long is the other parallel side? An irregular city block is divided into lots as shown. Find the area, in square feet, of lots numbered as follows: 44. Laying out public lands. In the more recently settled parts of the country land is laid out as here described. 45. Principal meridian. Through a given tract a meridian is chosen as the principal meridian. 46. Base line. An east and west line is chosen as the base line. The principal meridian and base line are here shown. X Base Principal eridian R. 3W. R. 2W. 'R. 1W. Line R. 1 E. 47. Townships. Lines are run parallel to the principal meridian and base line, at intervals of 6 mi. This divides the land into townships. 48. Ranges. The north and south rows of townships are called ranges. X on the first map is numbered T. 2 N., R. 3 W.; that is, the second township north of the base line, in the third range west of the meridian. N 654 321 7 8 9 10 11 12 49. Sections. A township is divided. into sections, each 1 mi. square. This map shows the method of numbering these sections. N.W. E. N.W. Each section is then divided as shown in the third map. If this map represents the shaded part Y of the second map, and that represents the shaded part X of the first map, the shaded part here shown would be thus described: S.W. of N.W., Sec. 21, T. 2 N., R. 3 W. This means the southwest quarter of the northwest quarter of section 21, second township north, third range west. OF N.E. OF N.W. 160 A 320 A In sections of the country where land is not laid out in this way little attention should be given to this subject. WRITTEN EXERCISE Write the description, plot, and find the area: 1. S.E., Sec. 5, T. 3 S., R. 3 W. 2. N.E., Sec. 8, T. 2 N., R. 2 W. 3. E. of N.W. 4, Sec. 2, T. 2 N., R. 3 E. 4. S. 5. N. of S.W. 1, Sec. 10, T. 3 S., R. 2 E. 6. E. 8. S.W. of N.W. 4, Sec. 32, T. 1 S., R. 3 E. 9. N.W. of N.W. 1, Sec. 6, T. 3 N., R. 3 W. 10. How much is this farm worth at $65 an acre: W. of S., Sec. 3, T. 2 N., R. 2 W.? 11. How much is this farm worth at $75 an acre: N.E. 4 of N.E. 4, Sec. 5, T. 2 S., R. 2 E.? 12. Find the area of this farm: S.W. 4, Sec. 10, T. 2 S., R. 2 E. Draw a plan of a township and locate the farm. 13. Find the area of this farm: N., Sec. 6, T. 1 N., R. 1 E. Draw a plan of a township and locate the farm. 14. Mr. Simmons owns the S. 1, N.E. 1, Sec. 3, T. 2 N., R. 3 E. How many rods of fence are needed to inclose it? 15. How far is it from Mr. Taylor's farm, N.W. of N.W. 4, Sec. 16, T. 1 N., R. 3 W., to Mr. Hunt's farm, S.W. of N.W., Sec. 28, T. 1 N., R. 3 W.? Draw the map. 16. A road running straight through a farm is mi. long and 3 rd. wide. How many acres in the road? If hay can be cut from the sides, averaging of the area, and the amount of hay averages 2 tons to the acre and is worth $9.50 a ton, what is gained by attending to this crop? II. MEASURES. PERCENTAGES. PROPORTION VOLUMES ORAL EXERCISE 1. Give the table of dry measure. 3. Give the table of liquid measure. 4. A cellar is 21'× 18', and 9' deep. Express the dimensions in yards; the volume in cubic yards. 5. At 50 ct. a cubic yard, how much will it cost to excavate a cellar containing 126 cu. yd.? 175 cu. yd.? 21' x 18' means 21 ft. by 18 ft. 6"× 5" means 6 in. by 5 in. 6. What is the volume of a box 7" x 3" × 8" ? 7. What is the volume of a box 6" x 5" × 10"? 8. How many cubic feet in a room 10' x 12' x 9'? 9. How many cubic feet in a room 12' x 12' × 10'? Find the volumes of solids of the following dimensions: |