(24) A haystack, 114 ft. high, has an oblong base 20 ft. long, and 8 ft. broad; the sides of the rectangular horizontal section 9 ft. from the ground through the eaves are 22 ft. and 8.8 ft.; the part above the eaves forms a triangular prism 22 ft. long; find the whole weight of the stack, if 200 cubic feet of the hay weigh

1 ton.

Ans. 9.154 tons.

(25) A hemispherical basin 25 ft. in diameter, 4 ft. deep, and ths filled with water, is drained by means of a 3 in. pipe, through which the water flows at an average rate of 2 miles per hour; shew that it will continue flowing for 2 hours 22 minutes.

MISCELLANEOUS EXERCISES.

(1) Find the area of a floor 31 ft. 9 in. long, and 18 ft. 7 in. broad. Ans. 590 sq. ft. 3 sq. in. (2) A square floor, whose side is 15 yds., is covered by 7200 equal square tiles; what is the length of a side of each tile? Ans. 6.363 inches.

(3) A chess-board having 8 squares along each side is 18 inches square. Find the length of a side of one of its squares. Ans. 24 inches.

(4) One hundred thousand men are drawn up in a square: how much space will they occupy, if to each man is allowed 2 ft. 3 in., by 1 ft. 9 in.*?

Ans. 43750 sq. yds.

(5) If the men in the last Ex. were drawn up in an oblong whose sides are in the proportion of 10 to 1, each man covering 2 ft. square, what would be the periphery of the oblong?

5

Ans. ths of a mile.

6

(6) Shew, without assuming any Rule, that the area of the rectangle, whose adjacent sides are 7ft., and 52 ft., is equal to 40 sq. ft.

* To avoid the frequent repetition of the words rectangular area it is usual, as here, simply to insert by between the numbers expressing the length and breadth of such an area.

(7) Shew, by a diagram, that 304 sq. yds. 1 sq. pole.

(8) Prove, by diagrams, that (3 ft.)2 = 1 sq. ft.; and that ft. ft. 16 sq. ft.

=

(9) Determine, by a diagram, how many equal squares, of 1 inches side, can be obtained from a rectangle 72 inches long, and 45 inches broad. Ans. 2560.

(10) A lawn is 70 yds. by 32 yds.; find the cost of laying it down with pieces of turf, each 15 in. by 9 in., at 10s. the gross. Ans. £74. 13s. 4d.

(11) A roof 45 ft. by 27 ft. is covered with slates, each 18 in. by 9 in. How many will be required?

Ans. 1080.

(12) The cost of paving a floor with flags, each 18 in. by 15 in., at 7d. per square foot, comes to £33. 9s. id.; how many flags were there in the floor? Ans. 576.

(13) A field of 74 acres is planted in rows at uniform distances of 15 inches; find the number of plants required for the whole field, if in each row the plants are half a yard apart. Ans. 174240.

(14) A field 40 poles by 24 poles is divided into 72 equal plots; find the number of square yards in each plot, and express the result as the decimal part of an (1) Ans. 4083. (2) Ans. ·084375.

acre.

(15) The walls of a room 8 yds. by 5 yds., and 11 ft. high, are painted at 9d. a square yard; what is the whole cost? Ans. £3. 14s. 3d.

(16) A straight road 45 ft. wide, and a furlong in length, is cut off the side of a field of 4 acres; how much is there left for cultivation ?

Ans. 3 ac. 1r. 1010 p. (17) What length must be cut off from a plank 9 inches wide, to make a door whose face is 16 sq. ft.? Ans. 21 ft.

(18) If the side of a square be 84 ft., what decimal Ans. .00625,

part of a rood is its area?

(19) The area of a square picture is 24 ft., and the width of the frame is 4 in.; how much wall does it cover? Ans. 43 sq. ft.

(20) From a square containing 1 acre there are subtracted 32 rectangular plots, each 12.6 yds. long, and 10.5 yds. broad; how much is left? Ans. 606 4 sq. yds.

(21) Twenty shutters 9 ft. high, are to be made to cover a shop window, the area of which is 40 sq. yds. 1 ft. 6 in. What must be the breadth of each shutter? Ans. 2.016 ft.

(22) Find the areas of the triangles whereof the sides are as follows:

(23) Given the following lengths of the sides and perpendiculars upon them from the centres of certain polygons; find their areas.

(24) Find the areas of the irregular polygons, of which the sides and diagonals are as follows.

[NOTE. The diagonals are all drawn from the extremity of that side whose measure stands first.]

(25) A tenant has £78. 2s. 6d. allowed him for draining a rectangular field by a channel traversing the diagonal; how much per lineal yard may he expend upon the drain without loss, if the sides of the field be 100 yards, and 75 yards? Ans. 12s. 6d.

(26) Find the relation between the sides of a rightangled triangle, whereof one of the acute angles_measures 30o. Ans. 1: √3: 2.

(27) The sides of a triangle are equimultiples of 3, 4, and 5; shew that its area is 6 times the square of the multiple.

(28) Shew that the ratio of the side of a square to its diagonal is 29: 41 nearly.

(29) A railway platform has two of its opposite sides parallel, and its other two sides equal; the parallel sides are 80 ft. and 92 ft. respectively; the equal sides are 10 ft. each; what is its area? Ans. 688 sq. ft.

(30) A field is bounded by four straight lines, of which two are parallel; if the sum of the parallel sides be 625 links, and their perpendicular distance be 160 links, what is the content of the field? Ans. 1 acre.

(31) A rectangular garden is to be cut from a rectangular field, so as to contain a quarter of an acre. One side of the field is taken for one side of the plot, and measures exactly 3.5 chains; how long must the other side be? Ans. Five-sevenths of a chain.

(32) The side of a rhombus is 10 ft., and the longer diagonal is 16 ft.; find the other diagonal, and the area of the rhombus. (1) Ans. 12 ft.; (2) Ans. 96 sq. ft.

(33) Upon the base of an equilateral triangle, whose side is 6 ft., another triangle is described, one-third of the original triangle in area, find its perpendicular height. Ans. 1.732 ft.

(34) An equilateral triangle has a perimeter of 375 links; find its area as a decimal part of an acre. Ans. 06765625.

(35) Find the cost of covering with asphalte, at 8d. per sq. yd., a triangular plot, whose sides are 40, 36, and 27.5 yds. Ans. £16. 14s. 8d.

(36) Find the area of the largest square which can be cut out of a circle whose radius is 1 foot.

Ans. 2 sq. ft. (37) The largest possible circle is cut out of an area of 15 ft. square; find the area of each of the corners remaining. Ans. 12 sq. ft.

(38) Two equal circles touch each other, and a cord tightly encloses them both without crossing itself; find the length of the cord, and the area enclosed by it, in terms of the radius.

(1) Ans. 10 rad. (2) Ans. 7×(rad.)2

(39) Two equal circles, of 1 inch radius, are distant 2 inches from each other, and a cord passes tightly round them, crossing between them, and in contact with twothirds of each circumference; find the length of the cord, and the area enclosed by it.

(1) Ans. 15.308 in. (2) Ans. 7.654 sq. in.

(40) Find how many circles of in. radius could be made from another of 1 foot radius, supposing the whole area of the larger circle could be used up. Ans. 576.

(41) If a pound's worth of silver, in sixpences, reaches 25 inches, when the coins are placed side by side in a straight line, what is the diameter of each coin, and the total surface covered by them?

(1) Ans. in. (2) Ans. 121 sq. in.

(42) The largest possible square is cut out of a given quadrant; compare the area of the square with that of the remainder of the quadrant. Ans. 7: 4.

(43) The corner of the leaf of a book is turned down twice, so that the lines of folding are parallel, and form, with the edges of the book, two similar rightangled triangles, whose heights are as 1 to 2; if the height and base of the smaller triangle are 2·5 in., and 1.75 in., respectively, find the area of the larger triangle. Ans. 174 sq. in.