Rule II. To find the diameter of a circle, multiply the circumference by .3183. Rule III. To find the area of a circle, multiply the circumference by one-fourth of the diameter, or multiply the square of the radius by 3.1416. WRITTEN EXERCISES. 1. The diameter of a circle is 12 feet; what is its circumference? SOLUTION. To find the circumference, we multiply the diameter by 3.1416; 3.1416×12 equals 39.27; hence the circumference equals 39.27 ft. 2. What is the circumference of the planet Venus, its diameter being about 7800 miles? Ans. 24504.48 miles. 3. The distance round a circular pond is 500 feet; what is the distance across the pond? Ans. 159.15 ft. 4. How many times will a carriage wheel 4 ft. 6 in. in circumference revolve in driving 10 miles? Ans. 11733. 5. I have a circular flower-bed 50 feet in circumference; what is the area of the bed? Ans. 198 sq. ft. 135 sq. in. 6. A cow is fastened to a stake by a rope 16 feet long; what space can she graze over? Ans. 89.36+ sq. yd. SUPPLEMENTARY PROBLEMS. To be omitted unless otherwise directed. 7. If the equatorial diameter of the earth is is 7925.75 miles, what are its circumference and the length of a degree of longitude at the equator? Ans. 24899.536+ miles; 69.16+miles. 8. A circular flower-bed being divided into four equal parts by lines drawn from the centre, one section was planted with tulips; what was the area of the tulip-bed, its outer edge being 7 feet? Ans. 15.5967 sq. ft. MEASUREMENT OF LAND. 344. The Unit of Measure of land is the Acre, which is sometimes divided into square rods and sometimes into square chains. Hundredths of an acre are also frequently used. Government lands are divided by parallels and meridians into townships, which contain 36 square miles or sections, and each section is sub divided into quarter-sections. Hence, 640 acres make a section, and 160 acres a quarter-section. The quarter-sections are still further subdivided into half-quarter-sections, quarter-quarter-sections, and lots. Lots are often of irregular form on account of natural boundaries, but contain, as near as may be, a quarter-quarter-section. NOTE. The pupil will remember that rods multiplied by rods give square rods, chains by chains give square chains; also, that 1 acre 10 square chains or 160 square rods. WRITTEN EXERCISES. 1. How many square rods in a grass plat 65 ft. long and 15 ft. wide? SOLUTION.-The area equals 65× 15, or 975 sq. ft.; reducing to square rods, we have 3 sq. rd. 2. How many acres in a rectangular meadow 725 rods long and 400 rods wide? Ans. 1812 A. 80 P. 3. What is the value of a farm 208.7 rods long and 120 rods wide, at $81 an acre? Ans. $12795.917. 4. Mr. A bought 64 A. 116 P. of land for $3.50 per square rod, and sold it for $3.75 per square rod; what did he gain? Ans. $2589. 5. A rectangular pond is 200 rd. 17 yd. long, and 150 rd. 15 yd. wide; required its area. Ans. 16 A. 14 P. 9.75 ch. wide; how 6. I have a field 16.5 ch. long and much land does it contain? 7. Mr. Wilson's farm contains 163 A. 3 ch., and its length is 71 ch.; how many rods of fence would surround it? be required to Ans. 752 rd. SUPPLEMENTARY PROBLEMS. To be omitted unless othermise directed. 8. If a township is equally divided among 480 families, how many acres does each family receive, and what part of a section does each receive? Ans. 48 acres; of a section. 9. How many rails are required to fence a quarter-quarter-section, the fence being 5 rails high, and each rail 8 ft. long; and what will be the cost at $35 per thousand rails? Ans. 3300 rails; $115.50. 10. A field 80 rods long contains 15 acres, while another field of the same width contains 9 acres; what is the length of the latter field? Ans. 48 rods. NOTE.-In Ex. 9, the area is 6400 sq. rd., and each side is 80 rd. 11. How much less will it cost to fence a field 72 rods square than a rectangular field 3 times as long and as wide, if fencing cost $2.50 a rod ? Ans. $480. 12. A mechanic having a lot of ground 50 rods square, planted 3 acres with corn, 200 square rods with vegetables, 15 rods square with flowers, and the remainder he kept to pasture his cow; how much of the lot was pasture? Ans. 9 A. 155 P. COST OF ARTIFICERS' WORK. 345. By Artificers' Work we mean plastering, painting, papering, paving, stone-cutting, etc. 346. Plastering, painting, papering, paving, and ceiling are estimated by the square foot or square yard. Roofing, flooring, partitioning, slating, etc., generally by the square, which consists of 100 square feet, but sometimes by the square foot or yard. 347. Shingles, which commonly measure 18in. by 4in. are estimated by the thousand or bundle. 1000 are generally allowed to a square of 100 sq. ft. WRITTEN EXERCISES. 1. What will be the expense of paving a sidewalk 303 ft. long and 7 ft. wide, at $2.25 per square yard? SOLUTION.-The area equals 303×71, or 22724 sq. ft., which equals 252 sq. yd.; hence the cost is $2.25×2521, or $568.12. 2. What will it cost to plaster a school-room 40 ft. long, 20 ft. wide, and 10 ft. high, at $0.36 a square yard? Ans. $80. 3. What is the cost of wainscoting a room 28 ft. long by 15 ft. 4 in. wide, to a height of 4 ft. 3 in. at $0.45 per square yard? Ans. $18.41. 4. What is the cost of slating a roof 52 ft. 10 in. long, each side being 20 ft. wide, at $15.25 per square? Ans. $322.28. wide, and 35 ft. 5. A frame house is 50 ft. long, 28 ft. high; what will be the expense of outside painting at $12.25 per square? Ans. $668.85. SUPPLEMENTARY PROBLEMS. To be omitted unless otherwise directed 6. What will it cost to shingle a roof 64 ft. long and 32 feet from eaves to ridge, the first course along the eaves being double, at $14.87 a thousand ? Ans. $614.992. 7. What will it cost to paper a room 40 ft. long, 32 ft. 4 in. wide, and 154 ft. high, allowing 815 sq. ft. for doors, windows and washboards, at 25¢ per sq. ft.; ceiling not included? Ans. $356.83. 8. A cistern 7 ft. 5 in. long, 4 ft. 6 in. wide, and 6 ft. 3 in. deep, is to be lined with zinc costing 124 a pound, allowing 5 lb. to the square foot; what will be the expense? Ans. $109.40. CARPETING AND PAPERING. 348. In Carpeting we take into consideration the width of the carpet, the allowance for matching the figures, and whether the strips run lengthwise or crosswise. To match the figures we must often turn under or cut off one of the ends. When an exact number of strips is a little too wide for the room, one breadth is turned under. Rule. Find the number of strips required, and multiply the number of yards in each strip by the number of strips. WRITTEN EXERCISES 1. How many yards of carpet, 1 yd. wide, will be required to carpet a floor 18 ft. 8 in. by 14 ft. 9 in., running lengthwise. SOLUTION.-The room is nearly 5 yd. wide, hence it will require 5 strips, or breadths, each 183 ft. long; and it will take 183 ft. X5= 93 ft., or 31 yds. 2. A lady wishes to cover her sitting-room, 31 ft. long and 15 ft. wide, with matting 2 ft. wide, no allowance required for matching; how many yards will it take, running lengthwise? How many running crosswise? Ans. 63 yds.; 65 yds. 3. How many yards of Brussels carpet, yd. wide, will it take to carpet a parlor 26 ft. long by 15 ft. wide, the strips running lengthwise, the matching of figures requiring 6 in. to be cut off each strip except the first? Ans. 613 yds. 4. Miss Hartman wishes to carpet (lengthwise) a room 18 ft. long by 14 ft. 6 in. wide, with Brussels carpet of a yard wide, at $1.25 a yard; what will it cost, 2 yards waste for matching? Ans. $62.50. SUPPLEMENTARY PROBLEMS, To be omitted unless otherwise directed. 5. How many rolls of paper 8 yds. long and 20 in. wide, will cover the walls and ceiling of a room 30 ft. long, 22 ft. wide and 10 ft. 8 in. high, deducting 142 sq. ft. for openings? Ans. 418 rolls. 6. What will be the cost of papering the above room at $2.40 a roll, putting also a gilt moulding around the top of the walls, at 12 cents a foot? Ans. $111.78. 7. A room contained 3 windows, which were curtained with brocatelle of a yard wide; 10 yards were required for each window @ $1.50, and the curtains were lined with silk of a yard wide @ $.87; how many yards of silk were required, and what was the whole cost of the curtains? Ans. 24 yd.; $66. MEASURES OF VOLUME. 349. A Volume is that which has length, breadth, and thickness or height. These three elements are called dimensions. A volume is also called a solid. 350. A Rectangular Volume or Solid is a volume bounded by six rectangles. The bounding rectangles are called faces. Cellars, boxes, rooms, etc., are examples of rectangular volumes. 3 feet high. 3 feet wide. 351. A Cube is a volume bounded by six equal squares. Or, a cube is a rectangular volume whose faces are all equal. 352. By the Contents or Solidity of a volume we mean the amount of space it contains. The contents are expressed by the number of times it contains a cube as a unit of meas ure. Rule I. To find the contents of a cube or rectangular volume, take the product of its length, breadth, and height. For, in the volume above, the number of cubic units on the base equals the length multiplied by the breadth, or 3×39, and the whole number |
