34.357, c = 36.72, A = 69° 20′, B 20° 40.5', b 17. The shadow cast on a horizontal plane by a vertical pole 10 ft. long is 15.5 ft. Find the angle of elevation of the sun. Ans. 32° 49.7'. 18. The angle of elevation of the top of a tree at a point 145.5 ft. from the tree and 5.5 ft. above its base is 51° 32'. Find the height of the tree. Ans. 188.63 ft. 19. A ladder 48 ft. long rests against a building and makes an angle of 75° 36′ with the ground. Find the distance it reaches up the building. Ans. 46 ft. 33° 14.8'. Find the 20. From the top of a tower 305 ft. high, the angle of depression of a man on the horizontal plane through the foot of the tower is distance the man is from the foot of the tower. 21. At 60 ft. from the base of a fir tree the angle of 75°. Find the height of the tree. Ans. 465.27 ft. elevation of the top is Ans. 224 ft. nearly. 22. What is the inclination from the vertical of the face of a wall having a batter of, that is, slants 1 ft. in a height of 8 ft.? Ans. 7° 7' 30". 23. What is the angle of slope of a road bed having a grade of 5 per cent? One with a grade of 0.25 per cent? (A road with a rise of 5 ft. in 100 ft. has a grade of 5 per cent.) Suggestion. Use S. and T. scheme. Ans. 2° 51′ 45′′; 0° 8' 36". 24. In surveying along the Lake Front in Chicago, a pile of coal was encountered. Measurements were taken as shown in Fig. 38. Find the distance on a straight line from A to E. Ans. 338.41 ft. 25. Locate the centers of the hcles B and C, Fig. 39, by finding the distance each is to the right and above the center O. The radius of the circle is 1.5 in. Compute correct to four decimals. Ans. B 1.2135 in., 0.8817 in.; C 0.4635 in., 1.4266 in. 26. In the parallelogram of Fig. 40, b = 17.6 in., c = 9.3 in., and 0 = 127° 25'. Find the value of the altitude h of the parallelogram. State its value as a formula in terms of c and 0. (See Art. 51.) Ans. 7.3863 in., h = c sin 0. b 27. A ladder 32 ft. long is resting against a wall at an angle of 72° 30'. If the foot of the ladder is drawn away 2 ft., how far down the wall will the top of the ladder fall? = Ans. 0.704 ft. 28. A man surveying a mine, measures a length AB 220 ft. due east with a dip of 6° 15′; then a length BC 325 ft. due south with a dip of 10° 45′. How much deeper is C than A? Ans. 84.57 ft. = 29. Find the number of square yards of cloth in a conical tent with a circular base, and the vertical angle 72°, the center pole being 12 ft. high. Ans. 45.14. 30. A wheel 4 ft. in diameter and an axle 6 in. in diameter and 5 ft. long are fastened rigidly together. If one end of the axle rests on the ground, find the inclination of the plane of the wheel to the ground. If the wheel and the axle are rolled along, find the radii of the circles formed by the end of the axle and the wheel. Ans. 9.08 in. and 72.65 in. How 31. In the side of a hill which slopes upward at an angle of 32°, a tunnel is bored sloping downwards at an angle of 12° 15' with the horizontal. far below the surface of the hill is a point 115 ft. down the tunnel? 32. Find the areas of the following isosceles triangles: Ans. 94.63 ft. (a) Altitude = 17 ft. and base angles each 49° 19′. Ans. 248.43 sq. ft. 23 ft. and vertical angle 47° 16'. = = Ans. 302.23 sq. ft. (b) Base (c) Each leg 18 ft. and base angles each 57° 17.5'. Ans. 147.32 sq. ft. 33. Find the radius of a circle circumscribed about a regular polygon of 128 sides if one side is 2 in. What is the difference between the circumference of the circle and the perimeter of the polygon? Ans. 40.747 in.; 0.02 in. 34. Find the areas of the following regular polygons with sides 6 ft.: (a) pentagon, (b) hexagon, (c) octagon, (d) decagon, (e) 12-gon, (f) 20-gon. Ans. 61.937 sq. ft.; 93.530 sq. ft.; 173.82 sq. ft.; 276.99 sq. ft.; 403.06 sq. ft.; 1136.5 sq. ft. 35. Find the difference in the areas of a regular hexagon and a regular octagon, each of perimeter 72 ft. Ans. 16.99 sq. ft. 36. If R is the radius of a circle, show that the area of a regular circumscribed polygon of n sides is given by the formula 37. Show that the area of a regular inscribed polygon of n sides is given by the formula Ans. 364.43 ft. 38. The radius of a circle is 62 ft. Find the perimeter of a regular inscribed pentagon. 39. The radius of a circle is 75 ft. Find the area of the regular inscribed octagon. Ans. 15,910 sq. ft. 40. Find the area of a regular dodecagon inscribed in a circle of radius 24 in. Ans. 12 sq. ft. 41. A circle 10 in. in diameter is suspended from a point and held in a horizontal position by 10 strings each 6 in. long and equally spaced around the circumference. Find the angle between two consecutive strings. Ans. 29° 50′ 42′′. 42. A girder to carry a bridge is in the form of a circular arc. The length of the span is 120 ft. and the height of the arch is 25 ft. Find the angle at the center of the circle such that its sides intercept the arc of the girder; and find the radius of the circle. Ans. 90° 28.7', 84.5 ft. 43. In a circle of 60 in. radius, find the area of a segment having an angle of 63° 15'. Find the length of the chord and the height of the segment; take of their product and compare with the area. Ans. 379.69 sq. in. 44. Find the area of the following segments by the approximate rule A = h3 2w hw, and by the exact method of trigonometry. Find the per cent of error by the approximate rule in each case. Here w stands for the width and h for the height of the segment. 45. A tree stands upon the same plane as a house whose height is 60 ft. The angle of elevation and depression of the top and base of the tree from the top of the house are 41° and 35° respectively. Find the height of the tree. Ans. 134.49 ft. 46. From a point 20 ft. above the surface of the water, the angle of elevation of the top of a tree standing at the edge of the water is 41° 15', while the angle of depression of its image in the water is 58° 45'. Find the height of the tree, and its horizontal distance from the point of observation. Ans. 65.50 ft.; 51.886 ft. 47. At a certain point the angle of elevation of a mountain peak is 44° 30′; at a distance of 3 miles further away in the same horizontal plane, its angle of elevation is 29° 45'. Find the distance of the top of the mountain above the horizontal plane, and the horizontal distance from the first point of observation to the peak. Ans. 4.0984 mi.; 4.1705 mi. 248 h Suggestion. Find two simultaneous equations involving the unknowns h and x representing the distances as shown in Fig. 41. These are tan 44° 30′ h and tan 29° 45' = 3+x Solve these algebraically for h and x. 48. At a certain point A the angle of elevation of a mountain peak is a; at a point B that is a miles further away in the same horizontal plane its angle of elevation is B. If h represents the distance the peak is above the plane and x the horizontal distance the peak is from A, derive the formulas: Note. In using these formulas, it is convenient to use natural functions. On page 120 is given a solution of the same problem, obtaining formulas adapted to logarithms. 49. Find the height of a tree if the angle of elevation of its top changes from 37° to 44° 30′ on walking toward it 80 ft. in a horizontal line through its base. Ans. 258.52 ft. 50. From a point A on the level with the base of a steeple, the angle of elevation of the top of the steeple is 42° 30'; from a point B 22 ft. directly over A, the angle of elevation of the top is 36° 45′. Find the height of the steeple and the distance of its base from A. Ans. 118.87 ft.; 129.72 ft. 51. A ship sailing east at 24 miles per hour observes a lighthouse at noon to be E. 26° N. At 12:30 the lighthouse is E. 39° N. Determine when the ship is nearest the lighthouse and what the distance is. Ans. 15 min. 26.1 sec. past 1 o'clock; 14.717 mi. 52. A ship sailing due north observes two lighthouses in a line due west; after an hour's sailing the bearings of the lighthouses are observed to be southwest and south-south-west. If the distance between the lighthouses is 8 miles, at what rate is the ship sailing? Ans. 13.66 mi. per hour. 53. The description in a deed runs as follows: "Beginning at a stone (A), at the N.W. corner of lot 401; thence east 112 ft. to a stone (B); thence S. 36° W. 100 ft.; thence west parallel with AB to the west line of said lot 401; thence north on west line of said lot to the place of beginning." Find the area of the land described. Ans. 6612.88 sq. ft. 37. Accuracy. It is of very great importance that one should bear in mind as far as possible the limitations as regards accuracy. The degree of accuracy that can be depended upon in a computation is limited by the accuracy of the tables of trigonometric functions and logarithms used, and by the data involved in the computation. The greater the number of decimal places in the table, the more accurately, in general, can the angles be determined from the natural or logarithmic functions; but, in a given table, the accuracy is greater the more rapidly the function is changing. Since the cosine of the angle changes slowly when the angle is near 0°, small angles should not be determined from the cosine. For a like reason, the sines should not be used when the angle is near 90°. The tangent and cotangent change more rapidly throughout the quadrant and so can be used for any angle. Most of the data used in problems are obtained from measurements made with instruments devised to determine those data more or less accurately. The inability to be precise in the data depends not only upon the instruments used, but upon the person making the measurements and upon the thing measured. A man in practical work uses instruments which are of such accuracy as to secure results suitable for his purpose. The data |