(42) How many houses can be built on a frontage of of a mile, each house having a frontage of 36 feet 8 inches, allowing for 11 streets, each having a width of 40 feet? (43) Find the cost of turfing a piece of ground 10 chains long and 5 chains broad, each turf being 15 in. long and 6 in. broad, and 100 turfs costing 1s. 3d. [1 chain=22 yds.] (44) Find the altitude of a triangle whose base is 75 and the other sides 65 and 20 yards long respectively. [First find the area by Art. 69, then double the area divided by the base will be the altitude or height required, Art. 68.] (45) What is the first difference of the cost of carpeting a room 13 feet by 15, with Kidderminster carpet, a yard wide, at 38. 9d. a yard, and Brussels carpet, of a yard wide, at 4s. 6d. a yard? If the latter will last 1 times as long as the former, which is the more economical in the end, taking no account of interest on money expended? [Area of floor 15 x 13195 sq ft. 213 sq. yds. Cost of Kidderminster carpet£1 1s. 3d., of Brussels carpet £6 108., difference £2 8s. 9d. If the latter will last 1 times the former, Brussels carpet costing £6 10s. is worth Kidderminster carpet costing £41s. 3d. x1.] (46) How much matting, 26 inches wide, will be needed to cover a room 26 feet long and 18 feet wide, having on one of the longer sides two recesses, each 7 ft. long and 4 ft. 4 in. wide, and on the opposite side two projections, each 8 ft. long and 13 inches wide? [Add the areas of the recesses to the area of the room, and subtract the areas of the projections.] (47) A man undertook to walk round a square field containing 13 acres 1089 square yards in 7 minutes. If his pace was at the rate of 5 miles an hour, by how much time did he win? [13 ac. 1089 sq. yds. = 64009 sq. yds. Side of square=sq. root of 64009== 253 yds. If he walk 5 miles an hour, he will walk a mile in 12 mins. 12 1760 mins. 6 mins. 54 secs. 1760 99 99 Hence time by which he won=6 secs. (48) How much paper yd. wide would be required for a room 25 ft. 3 in. long, 20 ft. 6 in. wide, and 18 ft. high? (49) The sides of a triangle are 51 ch., 85 ch., and 68 ch. respectively; find its acreage. (50) What is the length of one of the four equal sides of a rectangular park containing 98759-3476 sq. yds.? (51) The hypotenuse of a right-angled triangle is 617 ft., and one of the sides is 105 ft.; find the area in sq. ft. (29) Find the side of a square that shall be equal in area to a rectangle of which two adjacent sides are 32 and 2048. (30) The sides of a triangle are 39, 42, and 45 feet; find its area in square yards. (31) How many pieces of paper, 14 yards long and 18 inches wide, will paper a room 24 feet long, 18 feet wide, and 12 feet high? (32) Find the length of the longest straight line that can be drawn in the room in the last question. (33) Find the size of a chess board (on which are 64 squares) of which each square is 13 in. and the border 21 in. deep. (34) The painting of a hall 10 yds. long, 7 yds. broad, 4 yds. high, cost £38; what will be the cost of painting another hall 12 yds. long, 8 yds. broad, and 5 yds. high? (35) An oblong plot of ground is laid out for a row of 40 houses, a strip 36 feet wide is cut off to form a road. Each house is a square, and covers an area of 1681 sq. ft., the front garden is 20 feet deep, the back garden 40 feet deep; find the area of the whole plot in square feet. (36) A railway 11 yards wide is carried over a flat country for 1500 chains; the land costs £350 per acre, and the construction of the railway £26 per square chain; find the joint cost of these two items. [Note, 11 yds. = chain; 1 acre 10 sq. chains.] (37) A room is 36 feet long, 25 feet broad, and 18 feet high; how many pieces of paper will be required to cover it, each piece being 12 feet long and 18 inches wide? (38) How many yards of carpet 25 inches wide will cover a floor 19 ft. 7 in. long by 18 ft. 9 in. wide? Find the cost of the carpet at 5s. 6d. per yard. (39) An oblong plot contains 1107 sq. yds. 1 sq. ft. 2 sq. in. ; find the length of the sides, if one side is twice as long as the other. (40) Two fields are of equal area; one is 35 poles long and 25 poles broad, the other is 20 poles broad; give its length in terms of feet and inches. [5 yds. 1 pole.] (41) A rectangular public park contains an enclosed square plot, whose sides are parallel to those of the park, the adjacent sides being at distances of 400 feet and 600 feet respectively from the corresponding side of the park; the plot is enclosed with iron rails at 94d. per linear foot, and at a total cost of £253 6s. 8d., find the area of the park, in acres, yards, etc. [£253 6s. 8d.+9d.6400 no. linear feet in perimeter of plot. Side of plot =6400÷4=1600 ft. Length of park 1600+ 600+ 600 Breadth of park = 1600 + 400+ 400 2800 ft. 2400 ft.] (42) How many houses can be built on a frontage of of a mile, each house having a frontage of 36 feet 8 inches, allowing for 11 streets, each having a width of 40 feet? (43) Find the cost of turfing a piece of ground 10 chains long and 5 chains broad, each turf being 15 in. long and 6 in. broad, and 100 turfs costing 1s. 3d. [1 chain=22 yds.] (44) Find the altitude of a triangle whose base is 75 and the other sides 65 and 20 yards long respectively. [First find the area by Art. 69, then double the area divided by the base will be the altitude or height required, Art. 68.] (45) What is the first difference of the cost of carpeting a room 13 feet by 15, with Kidderminster carpet, a yard wide, at 3s. 9d. a yard, and Brussels carpet, of a yard wide, at 4s. 6d. a yard? If the latter will last 13 times as long as the former, which is the more economical in the end, taking no account of interest on money expended? [Area of floor 15 × 13195 sq ft. 213 sq. yds. Cost of Kidderminster carpet £1 1s. 3d., of Brussels carpet £6 10s., difference £2 8s. 9d. If the latter will last 1 times the former, Brussels carpet costing £6 10s. is worth Kidderminster carpet costing £1 1s. 3d. x1.] (46) How much matting, 26 inches wide, will be needed to cover a room 26 feet long and 18 feet wide, having on one of the longer sides two recesses, each 7 ft. long and 4 ft. 4 in. wide, and on the opposite side two projections, each 8 ft. long and 13 inches wide? [Add the areas of the recesses to the area of the room, and subtract the areas of the projections.] (47) A man undertook to walk round a square field containing 13 acres 1089 square yards in 7 minutes. If his pace was at the rate of 5 miles an hour, by how much time did he win? [13 ac. 1089 sq. yds. 64009 sq. yds. Side of square sq. root of 64009253 yds. If he walk 5 miles an hour, he will walk a mile in 12 mins. Hence time by which he won=6 secs. (48) How much paper yd. wide would be required for a room 25 ft. 3 in. long, 20 ft. 6 in. wide, and 18 ft. high? (49) The sides of a triangle are 51 ch., 85 ch., and 68 ch. respectively; find its acreage. (50) What is the length of one of the four equal sides of a rectangular park containing 98759 3476 sq. yds. ? (51) The hypotenuse of a right-angled triangle is 617 ft., and one of the sides is 105 ft.; find the area in sq. ft. (52) A school-room is 32 ft. long and 18 ft. wide; how many children will it accommodate, allowing 8 sq. ft. of floor for each child? (53) How many acres of land will be wanted for a road 4 miles long and 33 ft. broad? (54) The sides of a rectangle are 16 ft. and 24 ft. respectively; a border 2 ft. wide is taken off all round; find the remaining area. (55) The height of a wall is 36 ft.; at what distance from the base of it must the foot of a ladder 39 ft. long be placed so that it may just reach the top of it? (56) Supposing the cost of a carpet in a room 24 ft. long at 4s. 6d. per sq. yd. to be £12, find the breadth of the room. (57) A rectangular field measures 56 yds. in length, and a diagonal path across it is 65 yds. long; how many yards of fencing will be required to enclose it? (58) An oblong grass-plot 120 ft. by 60 ft. is to be levelled at £6 18. per sq. ch., and a lawn tennis court 78 ft. by 36 ft. is to be turfed within it at 4d. per sq. yd.; what will be the total cost? [481 sq. yds. = 1 sq. ch.] (59) A carpet contains 167.71545025 sq. ft.; find its breadth when it is 6 times longer than it is broad. (60) What is the area of a triangle whose sides are respectively 293, 234, and 85 ? (61) Two ships set sail from the same place at the same time; the one sails due west at the rate of 28 miles an hour, the other due south at 21 miles an hour: how far will they be distant from each other in 24 hours? (62) If a postage stamp be 1 in. long and in. broad, how many will be required to fill a single page of an album 16 in. square? (63) What is the altitude of a triangle whose area is 2 ac. 3 ro. 28 po., and the length of the base 1250 lks.? (64) There are two rectangular fields equal in area; the sides of one measure 216 yds. and 128 yds. respectively, and the longer side of the second is 192 yds.: what is the length of the shorter side? (65) What is the side of a square equal in area to a rcctangle whose adjacent sides are 32 yds. and 2048 yds. respectively? (66) The paving of a triangular court cost £18 7s. 6d.; the longest of the three sides was 126 ft., and the length of each of the other two equal sides was 105 ft.: find the price of paving per sq. yd. Butler & Tanner, The Selwood Printing Wo: ks, Frome, and London. PART II. MENSURATION OF PLANE SURFACES. (continued.) THE TRAPEZOID. 73. A Trapezoid is a quadrilateral or four-sided figure which has two sides parallel. ABCD is a trapezoid, of which A B, D C are the parallel sides, and D E or CF the perpendicular distance between them. D 74. To find the area of a trapezoid when the parallel sides and the perpendicular distance between them are given. RULE. Multiply half the sum of the two parallel sides by the perpendicular distance between them, and the product will be the area. If two straight lines are of unequal length, the average or mean length of the two is found by taking half the sum K HD C G L F B of their lengths. Thus, if two straight lines measure 10 and 12 inches respectively, the average or mean length is (10+12)=11 inches. Let G H or EF in the annexed figure represent the mean length of A B and CD, and EH or FG the perpen dicular distance, then it is evident that the area of the trapezoid ABCD is equal to the area of the 49 E |