is 24 feet broad, it is necessary to make a cutting 40 feet in depth; what must be the breadth of the cutting at top, supposing the slopes of the sides to be 65°? Ans. 61.305 feet. Prob. 8. The sides of a quadrilateral field are 690 yards, 467 yards, 359 yards, and 428 yards; also, the angle contained between the first and second sides is 57° 30', and the angle between the third and fourth sides 122° 30'. Required the area of the field. Ans. 200677.2 square yards. Prob. 9. There are two regular pentagons, one inscribed in a circle, and the other described about it; and the difference of the areas of the pentagons is 10 square inches. Required the radius of the circle. Ans. 8.926 inches. Prob. 10. What is the length of a chord cutting off one third part of a circle, whose diameter is 289 feet. Ans. 278.67 feet. Prob. 11. The area of a triangle is 1012; the length of the side a is to that of b as 4 to 3, and c is to b as 3 to 2. Required the length of the sides. Ans. a=52.470, b=39.353, c=59.029. Prob. 12. The area of a triangle is 144, the base is 24, and one of the angles at the base is 30°. Required the other sides of the triangle. Ans. 24 and 12.4233. Prob. 13. Seven men bought a grinding-stone of 60 inches diameter, each paying one seventh part of the expense. What part of the diameter must each grind down for his share? Ans. The 1st, 4.4508 inches; 2d, 4.8400 inches; 3d, 5.3535 inches; 4th, 6.0765 inches; 5th, 7.2079 inches; 6th, 9.3935 inches; 7th, 22.6778 inches. Prob. 14. The area of an equilateral triangle is 17 square feet and 83 square inches. What is the length of each side? Ans. 76.45 inches. Prob. 15. The parallel sides of a trapezoid are 20 and 12 feet, and the other sides are 15 and 17 feet. Required the area of the trapezoid. Ans. 240 square feet. Prob. 16. How many square yards of canvas are required to make a conical tent which is 20 feet in diameter and 12 feet high? Ans. 54.526 square yards. Prob. 17. The circumference of an hexagonal pillar is 7 feet, and the height 11 feet 2 inches. Required the solid contents of the pillar. Ans. 39.488 cubic feet. Prob. 18. The base of the great pyramid of Egypt is a square whose side measures 746 feet, and the altitude of the pyramid is 450 feet. Required the volume of the pyramid. Ans. 83,477,400 cubic feet. Prob. 19. A side of the base of a frustum of a square pyramid is 25 inches, a side of the top is 9 inches, and the height is 20 feet. Required the volume of the frustum. Ans. 43.102 cubic feet. Prob. 20. Three persons, having bought a sugar-loaf, would divide it equally among them by sections parallel to the base. It is required to find the altitude of each person's share, supposing the loaf to be a cone whose height is 20 inches. Ans. 13.8672, 3.6044, and 2.5284 inches. Prob. 21. If a cubical foot of brass were to be drawn into wire of one thirtieth of an inch in diameter, it is required to determine the length of the said wire, allowing no loss in the metal. Ans. 55003.94 yards; or 31 miles 443.94 yards. Prob. 22. How high above the surface of the earth must a person be raised to see one third of its surface? Ans. The height of its diameter. Prob. 23. If a heavy sphere, whose diameter is 4 inches, be let fall into a conical glass full of water, whose diameter is 5, and altitude 6 inches, it is required to determine how much water will run over. Ans. 26.272 cubic inches. Prob. 24. The capacity of a cylinder is a cubic feet, and its convex surface is b square feet. Required the dimensions of Prob. 25. A triangular pyramid, the sides of whose base are 13, 14, and 15 inches respectively, and whose altitude is 16 inches, is cut, at the distance of 2 inches from the vertex, by a plane parallel to the base. Required the volume of the frustum of the pyramid. Ans. 447.125 cubic inches. Prob. 26. The altitude of a cone is 10 inches, and the radius of its base is 5 inches. At what distance from the base must a plane pass parallel to the base, so as to cut off a frustum whose capacity is 20 cubic inches? Ans. 0.2614 inches. SURVEYING. Prob. 1. The angle of elevation of a spire I found to be 39° 27′, and going directly from it 225 feet on a horizontal plane, I found the angle to be only 24° 38'. What is the height of the spire, and the distance from its base to the second station? Ans. Height 238.02 feet, distance 508.18 feet. Prob. 2. Wishing to know the distance of an inaccessible object, I measured a horizontal base-line 1328 feet, and found the angles at the ends of this line were 84° 23′ and 43° 19′. What was the distance of the object from each end of the base-line? Ans. 1151.44 feet, and 1670.35 feet. Prob. 3. Wishing to know the distance between two inaccessible objects, C and D, I measured a base-line, AB, 3784 feet, and found the angie BAD=47° 32', the angle DAC-39° 53′, the angle ABC=46° 34', and the angle CBD=38° 1. What is the distance from C to D? Ans. 3257.36 feet. Prob. 4. Suppose a light-house built on the top of a rock; the distance between the place of observation and that part of the rock which is level with the eye, and directly under the building, is 1860 feet; the distance from the top of the rock to the place of observation is 2538 feet, and from the top of the building 2550 feet. Required the height of the light-house. Ans. 17 feet 7 inches. Prob. 5. At 85 feet distance from the bottom of a tower, standing on a horizontal plane, the angle of its elevation was found to be 52° 30′. Required the altitude of the tower. Ans. 110 feet. Prob. 6. At a certain station, the angle of elevation of an inaccessible tower was 26° 30'; but, measuring 225 feet in a direct line toward it, the angle was then found to be 51° 30′. Required the height of the tower, and its distance from the last station. Ans. Height 186 feet, distance 147 feet. Prob. 7. To find the distance of an inaccessible castle gate, I measured a line of 73 yards, and at each end of it took the angle of position of the object and the other end, and found the one to be 90°, and the other 61° 45'. Required the distance of the castle from each station. Ans. 135.8 yards, and 154.2 yards. Prob. 8. From the top of a tower 143 feet high, by the seaside, I observed that the angle of depression of a boat was 35°. What was its distance from the bottom of the tower? Ans. 204.22 feet. Prob. 9. I wanted to know the distance between two places, A and B, but could not meet with any station from whence I could see both objects. I measured a line CD=200 yards; from C the object A was visible, and from D the object B was visible, at each of which places I set up a pole. I also measured FC=200 yards, and DE=200 yards, and at F and E set up poles. I then measured the angle AFC-83°, ACF=54° 31′, ACD=53° 30', BDC=156° 25', BDE=54° 30', and BED= 88° 30'. Required the distance from A to B. Ans. 345.5 yards. Prob. 10. From the top of a light-house, the angle of depression of a ship at anchor was 3° 38′, and at the bottom of the light-house the angle of depression was 2° 43'. Required the horizontal distance of the vessel, and the height of the promontory above the level of the sea, the light-house being 85 feet high. Ans. Distance 5296.4 feet, height 251.3 feet. Prob. 11. An observer, seeing a cloud in the west, measured its angle of elevation, and found it to be 64°. A second observer, situated half a mile due east from the first station, and on the same horizontal plane, found its angle of elevation at the same moment of time to be only 35°. Required the perpendicular height of the cloud, and its distance from each observer. Ans. Perpendicular height 935.75 yards, distances 1041.1 and 1631.4 yards. Prob. 12. An observer, seeing a balloon in the north, measured its angle of elevation, and found it to be 36° 52'. A second observer, situated one mile due south from the first station, and on the same horizontal plane, found its angle of elevation at the same instant to be only 30° 58'. Required the perpendicular height of the balloon, and its distance from each observer. Ans. Perpendicular height 3.003 miles, distances 5.006 and 5.837 miles. Prob. 13. From a window near the bottom of a house which seemed to be on a level with the bottom of a steeple, I found the angle of elevation of the top of the steeple to be 40°; then from another window, 21 feet directly above the former, the like angle was 37° 30′. What was the height and distance of the steeple? Ans. Height 245.51 feet, distance 292.59 feet. Prob. 14. Wanting to know my distance from an inaccessible object, P, on the other side of a river, and having no instrument for taking angles, but a chain for measuring distances, from each of two stations, A and B, which were taken at 300 yards asunder, I measured in a direct line from the object P 60 yards, viz., AC and BD each equal to 60 yards; also, the diagonal AD measured 330 yards, and the diagonal BC 336 yards. What was the distance of the object P from each station A and B? Ans. AP-321.76 yards, BP-300.09 yards. Prob. 15. Having at a certain (unknown) distance taken the angle of elevation of a steeple, I advanced 60 yards nearer on level ground, and then observed the angle of elevation to be the complement of the former. Advancing 20 yards still nearer, the angle of elevation now appeared to be just double of the first. Required the altitude of the steeple. D B P Ans. 74.162 yards. Prob. 16. In a garrison there are three remarkable objects, A, B, C, whose distances from each other are known to be, AB 213, AC 424, and BC 262 yards. I am desirous of knowing my position and distance at a station, P, from which I observed the angle APB, 13° 30′, and the angle CPB, 29° 50 ́. Ans. AP 605.7122, BP-429.6814, CP-524.2365. |