brink of a river, and a point C, close to the bank of the river on the other side, is observed both from A and B ; the angle CAB is 52°, CBA 70°. Find the width of the river.

61. Two towers stand on a horizontal plane 144 ft. apart; a person standing at the foot of each tower observes the elevation of one tower to be double that of the other, but when he is halfway between them the angle of elevation of one tower is the complement of that of the other. Show that the heights of the towers are 108 feet and 48 feet respectively.

62. From the edge of one bank of a river, a person ascends 100 yards up a slope of 1 in 4, and observes the angle of depression of an object on the opposite bank close to the edge of the river to be 120. Find the breadth of the river.

CXVIII.

MENSURATION.

1. When two diagonals of a trapezium and the angle at which they intersect are given, show how to find its area. Ex. ABCD is a trapezium, AC=312·4, BD=612·5, the angle at which AC, BD intersect is 105° 20′. Find the area.

2. A regular triangular pyramid is contained by 4 equilateral triangles, the side of each triangle being 20 ft. Find the content of the pyramid.

3. AC is the diameter of a circle, and a diagonal of the inscribed quadrilateral ABCD. Given AB=30, BC=40, CD=10, find AD and the area of the figure.

4. A slice is cut from a sphere by two parallel planes : the distance between the planes is 4 feet, the radius of one circular section is 9 feet, and of the other 12 feet. Determine the radius of the sphere and the volume of the slice.

5. Two sides of a triangular field, the lengths of which

are 16 and 40 poles respectively, are inclined to each other at an obtuse angle. Find that angle, and the third side of the triangle, when the field contains exactly an acre.

6. The sides of a quadrilateral are 10, 7, 8, 9, and the angle between the second and third sides is 120°. Find the area of the figure.

7. The perpendicular height of the frustum of a cone is 7 feet, and the radii of the ends are 4 and 5 feet. Find its volume.

8. Find the whole surface and solid content of a square pyramid, each side of the base being 12 ft., and the slant height 25 ft.

9. The radius of a sphere is 10 ft. Find the volumes of the two segments into which it is divided by a plane, the perpendicular on which from the centre is 5 ft.

10. A pyramid has a square base, the area of which is 20-25 square feet; each of the edges of the pyramid passing through the vertex is 30 ft. Find the inclination of either of the triangular faces to the base, and determine by how much the volume of the pyramid differs from 3800 cubic feet.

11. How long will it take to fill a hemispherical tank of 6 ft. radius from a cistern which supplies, by a pipe, 6 gallons of water per minute, a gallon of water containing 277 27 cubic inches?

12. The frustum of a right cone is 6 ft. high, the radius of the smaller end is 2 ft., the radius of the larger end is 3 ft. Find its volume. Find the position of the plane parallel to the ends which will bisect the frustum.

13. What length of a gun of 9 inch bore will a charge of 15 lbs. of powder fill, if 30 cub. in. of powder weigh 1 lb. ?

14. A hemispherical bell of 10 ft. diameter is partially buried with its mouth downwards and in a horizontal position, so that only of the vertical radius of the bell appears above ground. What quantity of earth must be dug out in order to leave the bell entirely uncovered and surrounded by a cylindrical wall of earth?

15. If a cubic foot of metal weighs 4 cwt. 1 qr., and is worth 10 guineas a ton, what will be the cost of 1 mile of piping made out of it, with a 9-inch bore, and in. thick?

16. A solid sphere, whose radius is a, has a cylindrical hole bored through it, whose axis passes through the centre, and whose diameter is half that of the sphere. Determine the volume of the remaining portion.

17. The total length of the edges of a rectangular parallelopiped is 72 lineal feet, its entire surface measures 174 square feet, and its volume is 110 cubic feet. Find its length, breadth, and height.

18. A regular hexagon, each side of which is 10 ft., revolves about a line which joins the points of bisection of two opposite sides. Find the whole surface of the solid thus generated.

19. If the middle points of the adjacent sides of two similar trapeziums be joined, show that the figures thus formed are parallelograms, whose areas are in the same ratio as those of the trapeziums.

20. Two concentric circles have radii of 10 ft. and 15 ft. respectively. Calculate the area of the figure bounded by these circles and by radii inclined at an angle of 40° to each other.

21. If 30 cubic inches of powder weigh 1 lb., show that it will require nearly 9 lbs. to fill a shell whose internal diameter is 8 inches.

22. One side of a quadrilateral inscribed in a circle is 40 ft., the angles which it makes with its adjacent sides are 90° and 75° respectively; if the radius of the circle be 25 ft., find the area of the figure.

23. If π a2 (b) be given as the expression for finding

the content of the segment of a sphere, what do a and b represent?

24. What length of a gun 6 inches bore will be filled with 20 lbs of powder of which 30 cubic inches weigh 1 lb.?

25. One side of a rectangle is 6 in. Find the other side, when the area is equal to that of a trapezoid whose parallel sides are 7 in. and 8 in., the perpendicular distance between them being 12 in.

26. Regular hexagons are inscribed in circles whose radii are 3 and 5 in. What is the radius of the circle in which the area of the inscribed hexagon is a mean proportional between the areas of the other inscribed hexagons?

27. Define a pyramid. Find the content of a hexagonal pyramid; side of base 10 ft., perpendicular height 15 ft.

28. The edge of a cube is 12 in.; one of the angles of the cube is cut off, so that the part cut off forms a pyramid with each of its edges (terminating in the angle of the cube) 5 in. long. Find the volume of the remaining solid.

29. Find the number of cubic feet in a hexagonal room, each side of which is 20 ft. and its height 30 ft.; and which is finished above with a roof in the form of a hexagonal pyramid 15 ft. high.

30. What is the solid content of a sphere, when its surface is equal to that of a circle 4 ft. in diameter ?

31. Find the solid content of a log of timber 10 yds. 2 ft. 7 in. long, 2 ft. 11 in. broad, and 2 ft. 5 in. thick.

32. If a pressure of 15 lbs. to the square inch be applied to a circular plate 3 feet in diameter, what is the total pressure?

33. Find the area of the equilateral triangle circumscribing the circle whose radius is 60 ft.

34. Show how to find the area of a trapezoid.

35. Find the volume of a pyramid.

Ex.: A pentagonal pyramid, side of base 7.25 ft., height 12.6 ft.

36. The paving of a semicircular courtyard at 5s. a foot cost £20. Find the radius of the semicircle.

37. Find the volume of a segment of a sphere 7.5 in. high; the radius of the sphere being 12 in.

38. One side of a triangle is 200 yds., another is 88 yds. Determine the third side, when the area is 5095 square yds.

39. An inverted conical vessel, whose height is equal to the diameter of its rim (4 in.), is supported when filled with water; on the top of it is placed a heavy sphere with a diameter of 5 in. How much water will be left in the vessel ?

40. A quadrilateral has two of its sides parallel; these sides are 10 and 12 ft. respectively; the perpendicular distance between them is 4 feet. Find the area of the figure.

41. The perimeters of two similar trapeziums are as 3:7, a side of one is 3 in., its area 774 square inches. Find the area of the other.

42. A hemispherical basin holds 1 gal. of water. Find its diameter, if 1 gal.=277-27 cubic inches.

43. Show that the square described round a circle is equal to of the inscribed dodecagon.

44. The sides of a quadrilateral taken in order are 27, 36, 30, 25 ft.; the angle between the first two is 90°. Find its area.

45. Find the content of a right cone whose height is 210.6 ft., and radius of base 23 ft.

46. How many cubic feet of water are contained in a ditch shaped like the frustum of a wedge, 120 yds. long, 6 ft. deep, 10 yds. broad at the top and 4 at the bottom?

47. A ditch is 5000 ft. long, 9 ft. deep, 14 ft. broad at top, and 11 ft. broad at the bottom. How many cubic feet of water will fill it? If half that quantity of water is supplied, how high will it rise?

48. The base of a pyramid is a regular hexagon, each side =40 ft.; what must be its height that its cubical content may be the same as that of a sphere whose radius is 21.5 ft.?