(55) The perimeter of a rectangular field, whose length is three times the breadth, is 800 links. How many roods and poles does it contain? [If the length is three times the breadth, the perimeter will be eight times the breadth; and if the perimeter is 800 lks. the breadth will be 100 lks. and the length 300 lks.] 51. To find one side of a rectangle when the area and the other side are given. RULE. Divide the area by the given side, and the quotient will be the side required. Since the area of any rectangle is found by multiplying its length by its breadth, it follows that if any two of these be given, we can find the third: thus, If the length of a rectangle be 4, and the breadth 3, the area will be 12; or, Ex. What is the length of a paved courtyard containing 169 sq. ft. 39 sq. in., which is 10 ft. 5 in. broad? = By reduction, 169 sq. ft. 39 sq. in. = 24375 sq. in.; 10 ft. 5 in. - 125 in. Length area÷breadth=24375÷125-195 in. 16 ft. 3 in. [1. Both the area and the given side must be expressed in the same denomination. 2. The length of the side required will be expressed in the same denomination. 3. Areas are expressed in square measure, the lengths of sides in linear measure.] Ex. XVIII. Find the breadth in the following rectangles, having given the area and the length : (1) Area, 14661 sq. yds.; length, 543 yds. (2) Area, 18-04125 sq. yds.; length, 6.375 yds. (3) Area, 8 sq. yds. 6 sq. ft. 111 sq. in.; length, 16 ft. 7 in. (4) Area, 17 ac. 4156 sq. yds.; length, 294 yds. (5) Area, 23 ac. 1 ro. 30 po.; length, 1875 iks. (6) Area, 3 ac. 34 po.; length 12 ch. 85 lks, Find the length in the following rectangles, having given the area and the breadth : (7) Area, 16200 sq. yds.; breadth, 24 yds. (8) Area, 140 sq. yds.; breadth, 8 yds. (9) Area, 3 ac.; breadth, 66 yds. (10) Area, 2 ac. 18 po.; breadth, 325 Iks. (11) Find the breadth of a rectangular field of 16 ac. 2 ro. 10 po., whose length is 26 ch. 50 lks. (12) A rectangular common is 320 yds. wide, what length (in yds.) must be taken to enclose 40 ac.? (13) What must be the length of a plot of ground, if the breadth be 15 ft., that its area may contain 46 sq. yds.? (14) A plank is 16 in. broad; find what length must be cut off, that the area may be a sq. yd. (15) What length must be cut off a board 63 in. broad, that the area may contain 2 sq. ft. ? (16) What is the length of room 13 ft. 1 in. long, and which takes 17 sq. yds. 2 sq. ft. 131 sq. in. of carpet to cover it? (17) The expense of paving a street half a mile long, at 71d. per sq. yd. was £440, find the breadth of the street. (18) There are two rectangular fields equal in area. The sides of the first are 1728 lks. and 441 lks., and one side of the second is 252 lks.; find the other side. (19) A square garden 256 yds. long is exchanged for an oblong of the same area. The length of the rectangular garden is 512 yds., find the breadth. (20) Two fields, one oblong and the other square, are measured and found to contain exactly the same area. The length of the square is 24 yds., and the length of the oblong 32 yds., find the breadth of the oblong. (21) A room 39 ft. long requires 26 sq. yds. of carpet to cover it; what is the breadth of the room? (22) A rectangular field contains 33 ac., and is 125 yds. wide; find its length. (23) A rectangular garden is to be cut off from a rectangular field, so as to contain a quarter of an acre. One side of the field is taken for a side of the plot, and measures 2 chains; find the length (in links) of the other side. 52. To find the length of carpet required to cover the floor of a room. RULE. Divide the area of the room by the width of the carpet, and the quotient will be the length required. Ex. How many yards of carpet 27 in. wide will be required to cover the floor of a room 18 ft. long and 12 ft, broad? Length of carpet required x width of carpet = area of floor. .. Length of carpet required area of floor÷width of carpet. Now, area of floor = 18 x 12=216 sq. ft.=31104 sq. in. Width of carpet=27 in. = .. Length of carpet required=31104÷27=1152 in. =96 ft. = 32 yds. = [1. The area of the room and the width of the carpet must be both expressed in the same denomination. 2. The length of carpet required will be in the same denomination. 3. The length and width of the carpet are both expressed in linear measure, the area of the floor in square measure. 4. The cost may be found, as before, by Practice.] 53. In a similar manner, the width of the carpet, the length of the room, the breadth of the room, and the price of the carpet per yard, may each be found when the other quantities are given. Ex. XIX. (1) What length of carpet 2 ft. wide, will cover a room 18 ft. by 16 ft.? (2) How many yards of carpet 27 in. wide will be required for a room 54 ft. by 15 ft.? (3) How many yards of matting 24 ft. wide will cover a floor that is 293 ft. long and 26 ft. broad? (4) How many yards of matting 2 ft. 6 in. wide would cover the floor of a room whose length is 27 ft. and breadth 20 ft.? (5) What length of carpet that is 3 ft. 9 in. broad will be equivalent to 37 ft. 9 in. in length and 7 ft. 6 in. in breadth? (6) A room is 26 ft. 3 in. long and 15 ft. 9 in. broad. Find the cost of covering it with carpet which is three-quarters of a yard wide at 4s. 6d. per yard. = [1 yd. 36 in., of a yd. = 9 in., of a yd. 27 in.] (7) What length of carpet of a yard wide will cover a room 36 ft. long and 27 ft. 9 in. wide; and what will be the cost of the carpet at 4s. 9d. a yard? (8) Find the expense of carpeting a room 12 ft. 6 in. long by 12 ft. 3 in. broad with carpet yd. wide, at 2s. 3d. a yard. (9) What length of carpet half a yard wide would be required for a room 27 ft. long and 17 ft. broad, and what would be the cost at 5s. 3d. a yard? (10) Find the expense of carpeting a room 18 ft. 9 in. long and 17 ft. 6 in. broad with carpet 30 in. wide at 5s. 9d. per yd. (11) What is the expense of carpeting a room 28 ft. long and 21 ft. wide with carpet 2 ft. 4 in. wide at 5s. 9d. per yd.? (12) The dimensions of a room are 29 ft. by 11 ft. What length of carpet § yd. wide will cover it, and what will be the expense at 33s. per yd.? (13) If 64 yds. of carpet 2 ft. wide will cover a room 24 ft. long, find the breadth of the room. [Area of room-length of carpet x width of carpet. (14) If 69 yds. of carpet 3 qrs. wide cover a room 8 yds. 2 qrs. 2 nls. long, find the breadth of the room. (15) If 46 yds. 2 ft. of matting 27 in. wide are required for a room 21 ft. long, find the breadth of the room. (16) If 51 yds. of drugget 2 ft. wide will cover a room 17 ft. broad, find the length of the room. = [Area of room length of carpet x width of carpet. Length of room area of room ÷ breadth of room.] (17) A room is 16 ft. wide. It requires 32 yds. of drugget a yard wide to cover it; what is the length of the room? (18) If 72 yds. of carpet will cover a room 15 ft. by 12 ft., what is the width of the carpet? [Area of room length of room x breadth of room. area of room + length of carpet.] (19) If 58 yds. of carpet will cover a room 27 ft. long and 14 ft. 6 in. broad, find the width of the carpet. (20) If 67 yds. of carpet will cover a room 22 ft. 4 in. by 20 ft. 3 in., what is the width of the carpet? (21) The cost of carpeting a room 21 ft. long with carpet 2 ft. wide at 5s. per yd., is £14; find the breadth of the room. [Length of carpet in yds. total cost cost per yd. Area of room length of carpet x width of carpet. (22) The cost of carpeting a room 18 ft. long with carpet 30 in. wide, and worth 5s. a yd., is £8 8s.; what is the breadth of the room? (23) The expense of carpeting a room 14 ft. broad with carpet 24 in. wide, at 4s. 6d. per yd., is £7 17s. 6d. ; required the length of the room. [Length of carpet in yds. = total cost ÷ cost per yd. Area of room = length of carpet x width of carpet. Length of room area of room breadth of room.] (24) A room is 10 yds. 2 ft. long and 7 yds. 1 ft. broad. The cost of covering it with carpet worth 4s. 6d. a yd. is £23 9s. 4d.; find the width of the carpet. [Area of room length of room x breadth of room. Length of carpet in yds. = total cost ÷ cost per yd. Width of carpet area of room ÷ length of carpet.] (25) The cost of a yard wide, is £15. by 4 yds.; find the covering a room with Brussels carpet, half The dimensions of the room are 15 yds. price of the carpet per yd. [Area of room = length of room x Length of carpet area of room Price of carpet per yd. = total cost breadth of room. length of carpet in yds.] (26) A person intended to buy a Brussels carpet yd. wide, price 48. 6d. per yd., for a room 40 ft. by 28 ft., but afterwards decided to put down a square Turkey carpet, each side 20 ft., in the middle, and a drugget, a yard wide, in the remainder of the room. The Turkey carpet cost £25. Find the cost of the drugget per yd. so that the whole expense may be the same as for the Brussels carpet. [Area of room = 40×28 = 1120 sq. ft. Cost of Brussels carpet = £37 6s. 8d. Portion covered by Turkey carpet 20 x 20 = 400 sq. ft. Portion to be covered with drugget = 1120-400720 sq. ft. =80 sq.yds. Cost of drugget £37 6s. 8d. - £25 £12 6s. 8d. Cost of drugget per yd. = £12 6s. 8d. ÷ 80.] (27) The sum of £8 11s. is spent upon the floor of a room 24 ft. long and 18 ft. broad; the centre of the room is covered with carpet 2 ft. wide at 4s. 3d. per yd., leaving a margin of 3 ft. all round the carpet. How much per sq. ft. does the margin cost to paint? [Area of room 24×18=432 sq. ft. Area of portion covered by carpet 18 x 12 = 216 sq. ft. Area of margin = 432 - 216 216 sq. ft. Cost of carpet = £7 13s., cost of painting = £8 11s.-£7 13s. = 18s.] 54. To find the area of the surface of the walls of a room. RULE. Multiply the distance round the room by the height. The four walls of a room may be represented by the four sides of a small rectangular box. The depth of the box will represent the height of the room, the longer side the length, and the shorter side, the breadth. If the box be taken to pieces and the four sides laid end to end, as in the annexed diagram, it will be seen that they form a rectangle, whose length, or base, is equal to the sum of the lengths of the four sides, whose breadth is the depth of the box, and whose area is the product of these quantitics. But these represent the distance round |