60. When a board is six inches wide, how long must it be to con tain 1 sq. ft.?-3 sq. feet?-7 sq. ft. ?-12 sq. ft.? A. Total 46ft. 61. If a road 150 miles long and 4 rods wide, would cost, when completed, $2 per square rod, what would the land cost by the acre, allowing the cost of making the road to be $2 per rod, (linear measure)? A. $240 per acre. 62. Suppose a square has an area of 7500 square yards; what is the breadth of a walk round it that shall take up just two thirds of the square? A. 18.3013yd. nearly. 63. If a rhomboid is 50 feet long and 40 feet wide, what is its area? A. 200ft. 64. If one rhombus be 60 feet long, with a breadth of 15 feet, and another 45 feet long, with a breadth of 20 feet, what is the difference in their areas? A. Nothing. 65. If a rhomboid be 80 feet long and 60 feet wide, what is the sum of the areas of the two ends which when cut off will leave the remainder in the shape of a square? A. 1,200 feet. 66. Herodotus estimated the largest and most remarkable of the Egyptian pyramids to be 800 feet square at its base. Now, how long road 4 rods wide would occupy as much land as the base of the pyramid? A. 1m. 6fur. 27-757rd. 67. Suppose the hypothenuse and perpendicular of a right angled triangle be 50 and 30 feet, what is the area? A. 600 feet. 68. If the sides of an oblique angled triangle be 40, 50, and 80 feet, what is the area? A. 818+ sq. ft. 69. If the sides of a triangle be 16.6; 18.32, and 28.6, what is its area? A. 143 nearly. 70. Suppose a field has one right angle, and its hypothenuse and base are 100 and 80 rods; how many acres does it contain? A. 15A. 71. Suppose a piece of land in the form of a right angled triangle, whose angles are respectively 120 and 160 rods: what is the area? A. 60A. 72. What is the circumference of a circle whose diameter is 15? (15×355÷113.) A. 47.12+. 73. What is the diameter of a circle whose circumference is 350? A. 111.4 +. 74. What is the area of a circle whose diameter is 24, and circumference 75 ? A. 450. 75. What is the area of a circle whose diameter is 24? For the method of solving several questions, see reference from 49, above. A. 452.3904+. 76. What is the area of a circle whose circumference is 75? A. 447.61875+. 77. If the diameter of a circle be 24, what is the length of one side of a square equal to the circle? A. 21.269+. 78. When the circuinference of a circle is 75, what is the side of a square equal to the circle? A. 21.157+. 79. What is the diameter of a circle whose area is 115? A. 12.1+. 80. When the side of a square is 10.5 what is the diameter of a circle which is equal to the square? A. 11.847+. 81. When the side of a square is 10.5 what is the circumference of a circle equal to the square? A. 37.224+. 82. When the diameter of a circle is 12, what is the area of a semicircle formed from that circle? A. 56.548+. 83. Suppose a tract of land is 5 miles long and 3 miles wide, what is he distance round a square of an equal area? A. 15m. 3fur. 373rd. 84. If a field be 48 rods long and 10 rods wide, what will be the diameter of a circle of equal area? Having found the area of a cir cle, find the sum of the areas of a square inscribed, and one circum scribed;* also the side of a triangle inscribed. A. Diameter, 24.721+ rods; areas, 916.635 + sq. rd.; side, 21.408+rd. 85. How long will it take a man, going at the rate of 10 miles in 2 hours, to travel round an area of 256,000 acres, laid out so that the circumference shall be the shortest distance possible that will contain the given area? A. 14h. 10,97m. 86. A circle has an area of 308 square rods, and is to be divided into four equal concentric circles; what will be the width of each circular part? A. 9.9+rd.; 2.05+rd.; 1.57+rd.; 1.33+rd. 87. The radii of two concentric circles are 10 and 12 yaids. What is the area included between them? A. 138.22996+yd. 88. There is a meadow of 10 acres in the form of a square, and a horse tied equidistant from each angle or corner. What must be the length of the rope that will permit the horse to graze over every part of the meadow? A. 28.284+rd. 89. In the midst of a meadow well stored with grass, I've taken just two acres to tether my ass; 90. What is the perimeter of Fig. 18, and what the area of its lots, a and b? A. Perimeter, 101.4159rd.; a=300 sq. rd.; b= 150 sq. rd. 91. Since an acre is equal to a rectangle, which is 40 poles= 10 chains 1,000 links in length, and 4 poles 1 chain=100 links in breadth, it will contain 1,000 × 100 100,000 square links, therefore = A D FIG. 18. 10 rods. a Area. 157.079 5 sq. rd. b 25 rods. B 92. If the linear dimensions be expressed in links, ana the superfi The diameter of the circle is of course the length of one side of a circumscribed square. 1 CONCENTRIC, [L. concentrico.] Having a common centre. 180501 cial contents be found, these results, when divided by 100,000, or with five figures cut off towards the right, will give the number of acres ana parts of an acre, expressed in decimals. 93. The length of a rectangular field being 25 chains 8 links, and its breadth 14 chains 75 links, what number of acres does it contain? 25 chains 8 links=2,508 links, and 14 chains 75 links=1,475 links; then 2,508 × 1,475=36.99300 acres=36 acres 3 roods 38.88 poles. A. 36A. 3r. 3823rd. 94. Find the area of a square field whose side is 10 chains. A. 11A. 4rd. 95. The base of a triangular field is 16 chains 3 poles, and its perpendicular 6 chains 2 poles; what number of acres does it contain? A. 5A. 1R. 31rd. 96. What is the length of the side of a square field comprising 2 acres and 4 poles? A. 4 chains. 97. Two acres of land are to be cut from a rectangular field whose breadth is 2 chains 50 links, by a line parallel with either end; what is the length of the plot? A. 8 chains. FIG. 19. 98. In the foregoing plot, the figures on the sides of each lot represent so many chains and links. The road that extends round the whole and terminates at the inner circle is 1 chain in width. Required with these data the number of acres that are contained in the whole figure? Answers.-a=1129.375c.; b=2,349.5c.; c=2,874 .4375c.; d= 2,085c.; e=1,077.625c.; f=1,271.875c.; g=2,945.59c.; h=2,475. 9375c.; i=3,214.375c,; j=829.576c.; k=5,690.3125c.; square without the circles=1,036.584c. Total, 2,698A. 3sq.rd.; without the road=181⁄2 chains, nearly. CXII. 1. A SOLID is any thing that has three dimensions, length, breadth, and thickness. 2. A PRISM is a solid with two equal and parallel bases or ends, and sides that are parallelograms. The sides are called the latera surfaces. 3. Prisms receive particular names, according to the figure of their bases, as triangular, circular, square, pentagonal, and so on. 4. A CUBE is a solid or prism bounded by six equal and square sides. 5. A PARALLELOPIRED is a prism of four sides and two ends, whose length is more than its breadth, as a hewn stick of timber. 6. A CYLINDER is a round prism, whose bases or ends are of course circular, like a round column, or a common round rule. 7. A PYRAMID is a solid whose sides taper gradually from the base to one common point, called the vertex of the pyramid. 8. Pyramids take their names according to the figure of their bases or ends, as circular, triangular, square, and so on. 9. A CONE is a round pyramid; of course its base is circular, as a sugar loaf, if it comes to a point at the top. 10. A SPHERE or GLOBE is a round, solid body, that has a centre equally distant from every part of the surface, as an orange. 11. The DIAMETER and PERIPHERY of a sphere are the same as those of a circle of equal circumference. A HEMISPHERE is half a globe. 12. A FRUSTRUM or trunk of a pyramid is a portion of the solid next the base, cut off so that its bases are parallel. The other part is called a segment. 13. Thus the top of a sugar loaf of pyramidal form, cut off square, is a segment, and what remains is a frustrum. 14. The Axis of a solid is a straight line passing from one end to the other, through the centre. 15. The Axis of a sphere is the same as the diameter of a circle. 16. The ALTITUDE or HEIGHT of a pyramid is the perpendicular distance from the apex or top to the centre of the base. 17. The Slant height of a regular pyramid is the distance from the vertex to the middle of one of the sides of the base, or, if it be a cone, to the circumference of the base. 18. A WEDGE is a solid that has a rectangular base, two triangular sides, and two quadrilateral sides that meet in an edge, as the wedge used in splitting wood. 19. A PRISMOID differs from a prism or a frustrum of a pyramid only in having its ends dissimilar. RULES FOR FINDING THE AREAS AND CONTENTS OF SOLIDS. 20. To find the content of a cube.-Cube either side. CXII. Q. What is a solid? 1. Prism? 2. Parallelopiped? 5. Cylinder? 6. Pyramid? Sphere? 10. Diameter? 11. Frustrum? 12. of a solid? 16. Wedge? 18. Prismoid? 19. solidity of a cube? 20. 7. Cube? Their names? 3. 21. To find the solidity of a prism, or cylinder, as of round timber.Multiply the area of its base by its length.* 22. To find the area of a prism or cylinder.-Add together the areas of the different sides and ends. 23. To find the solidity of a parallelopiped, as of square timber.Multiply the length by the breadth, and that product by the depth. 24. To find the solidity of a pyramid.-Multiply the area of the ase by of its height. 1 25. To find the area of a pyramid.-Multiply half the slant height by the perimeter of the base for the lateral surface, to which add the area of the base. 26. To find the solidity of a sphere.-Multiply the cube of the diameter by .523€ Or multiply the square of the diameter by of the circumference. Or multiply the surface by of the diameter. 27. To find the area or surface of a sphere.-Multiply the diameter by the circumference. 28. To find the solidity of a frustrum of a pyramid.-Add together he areas of the two ends, and the mean proportional between these areas; then multiply the sum by of the perpendicular height. 29. To find the area of a frustrum of a pyramid.-Add together the areas of the sides and ends. 30. To find the solidity of a wedge.-Multiply half its length into the length and breadth of its base. 31. To find how large a cube may be inscribed in a given sphere, or be cut from it.-Divide the square of the diameter of the sphere by 3, and extract the square root of the qaotient. 32. The side of a cube is 24 feet. Required its content. A. 13,824 feet. 33. When the side of a cube is 25.5 inches, what is the solidity? A. 16,581.375 inches. 34. A prism is 203 feet high, with a base 24 feet square. Required its content. 35. A stick of timber is 20 feet long, 1 foot 8 inches broad, and 10 inches thick. Required its solidity. A. 277 feet. A. 104.625. Q. Of a cylinder? 21. Of square timber? 23. Of a pyramid? 24. Of & sphere? 26. Of a frustrum of a pyramid? 28. Of a wedge? 30. The dimensions of round timber are found by girting the tree and taking of the girt for the side of a square. When the tree tapers regularly, the girt may be taken at the middle, or at both ends, in which case, half their sum will be the mean girt. When the timber is very irregular, the girt may be taken at several places, equally distant, and their sum divided by the number of girts; or divide the tree into several lengths, according to its irregularity, find the content of each length separately, and their sum will be the whole content of the tree. In measuring oak timber with the bark on, a deduction of or of the circumference is often made to the buyer; in respect to elm, beech, ash, &c. the deduction is less, because the bark is not so thick. GENERAL RULE.-Multiply the square of the quarter girt, or the square of of the mean circumference, by the length of the timber. This method, though very generally used, gives the content about less than that found by considering the tree as a cylinder, or the content will be nearly the same as it the tree were hewn square. |