To find the solidity of a pyramid. Definition. A pyramid is a solid, whose sides are all triangles, meeting in a point at the vertex, and the base any plane figure whatever. RULE. Pyramid. Multiply the area of the base by the altitude, and divide the product by 3. 1. Required the solidity of a pyramid, of which the area of the base is 95 and the altitude 15. Ans. 95 x 15-1425-3=475. 2. What is the solidity of a pyramid, the area of whose base is 403 and the altitude 30? Ans. 4030. 3. Required the solidity of a triangular pyramid, whose height is 30, and each side of the base 3? Thus: 32.433013=3.897117 area of the base, then 3.897117 X30 38.97117 solidity required. 4. A pyramid with a square base, of which each side is 30, has an altitude of 20; what is the solid content? To measure a parallelopipedon. Definition. A parallelopipedon is a solid of three dimensions, length, breadth, and thickness; Ans. 6000. as a piece of timber exactly square, whose length Parallelopipedon. is more than the breadth and thickness; the ends are called bases, which are equal. The solidity of a parallelopiped is equal to the product of the base into the perpendicular altitude. And a parallelopiped and a cylinder, which have equal bases and altitudes, are equal to each other. Find the area of the base, then multiply that by the length, and it will give the solid content. 1. If the side of a stick of timber is 1.75 feet, and the length 9.5 feet, to find the content. Thus: 1.7523 0625-area of base 9.5=29.09375 content. Ans. 2. A vessel 3.5 feet each side within, and 5 feet deep, to find the content. Ans. 3.5 3.5 12.25 × 5=61.25 content. 3. A piece of timber is 1 foot 6 inches broad, 9 inches thick. 9 feet 6 inches long; required the content. Ans. 1.5 X.75 1.125 × 9.5 10.6875 content. 4. What is the solidity of a cylinder whose height is 121 and diameter 45.2? Ans. 45.22 X .7854 × 121-192442.6. 5. The Winchester bushel is a hollow cylinder 18 inches in diameter, and 8 inches deep; what is its capacity? Ans. 18.52.7853982=268.8025 x 82150.42 cubic inches 6. There is a cistern under ground, in the form of a parallel ogram, in length 16 feet, width 12 feet, depth 9 feet; required the number of hogsheads it will contain. Thus: 16×12×9= 1728 cubic feet of space, then 17282=2985984 cubical inches, which divide by 231, the number of inches in a gallon=12926 gallons 63 205 hogsheads, 11 gallons. Ans. 7. How many hogsheads will a circular cistern contain which is 7 feet in diameter, and 9 feet in depth? Ans. 41 hogsheads, 7 gallons, 3 quarts, I pint. When the breadth and thickness of a piece of timber are given in inches, to find how much in length will make a solid foot. RULE. Divide 1728 by the product of the breadth and depth, and the quotient will be the answer or length, making a solid foot. In a piece of timber 11 inches broad and 8 inches thick, how many inches in length will make a solid foot? Ans. 11x8=88)1728(19.6. To find how many solid feet a round stick of timber of equal thickness will contain, when hewn square. RULE. Multiply twice the square of its radius, in inches by the length in feet, then divide the product by 144, and the quotient will be the answer. 1. Admitting the radius of a round stick of timber to be 11 inches, and its length 20 feet, how many solid feet will it contain when hewn square? Thus: 11×11×2×20=4840÷144 =33.6 solid feet of hewn timber. To find how many feet of square-edged boards of a given thickness, can be sawn from a log of a given diameter. RULE. Find the solid content of the log when made square by the last rule; then say, as the thickness of the board, including the saw-calf, is to the solid foot :: so is 12 inches to the number of feet of boards. 1. How many feet of square-edged boards 1.25 inches thick, including the saw-calf, can be sawn from a log 20 feet long and 24 inches in diameter ? Thus: 12 x 12 x2=288×20=5760 ÷144=40 feet solid content; then, 1.25: 40 :: 12: 384 feet of boards, Ans. Miscellaneous Matter in Mensuration. 1. Purchased a box of window-glass containing 256 panes of glass, 8 by 10 inches; required the number of square feet, and the cost at 5c. per foot. Ans. 8×1080 in. : 256 × 80=20480÷144-142.22 square feet, D7.11.1. 2. Bought a box of window-glass containing 320 square feet of glass 10×12 inches; required the number of panes. Ans. 10×12 120: 320 × 144-46080120-384 panes. 3. In a pile of wood 7.5 feet long, 61 feet high, and cut 4.75. feet, how many cords, and how much will it come to at D5 per cord? Ans. 7.5 x 6.25 × 4.75=22265625÷128=1.73950 cords x D5 8.69.75. 4. From a mahogany plank 26 inches broad, a yard and a half is to be sawn off; what distance from the end must the line be struck? Ans. 6.23 feet. 5. A joist is 8 inches deep, 3.5 broad; what will be the dimensions of a scantling just as large again as the joist, that is 43 inches broad? Ans. 12.52 inches deep. 6. How many 3-inch cubes can be cut out of a 12-inch cube? Ans. 64. 7. A may-pole, whose top was broken off by a blast of wind, struck the ground at 15 feet distance from the bottom of the pole; what was the height of the whole may-pole, supposing the length of the broken piece to be 39 feet? Ans. 75 feet. 8. The diameter of a circle is 25 rods; required the length of a stone wall that will enclose it. Ans. 78.54 rods. 9. How many men may stand on one acre of land, allowing 12 square feet to each man? 10. How many solid yards of earth will be make a cellar 48 feet long, 27 wide, and 6 deep? Ans. 3630. thrown out to Ans. 288. 11. There is an island 50 miles in circumference, and three men start together to travel the same way about it; A. goes 7 miles per day, B. 8, and C. 9; when will they come together again? Thus, 50x750x8+50x9÷7+8+9= 50 days, time of meeting, and A. will travel 350 miles, B. 400, C. 450. Ans. 12. A general disposing his army into a square battalion, found he had 231 men remaining, but increasing each side with one soldier, he wanted 44 to fill the square; of how many men did his army consist? Thus: 231+44=275 and 275-1-2-137, then 137 × 137+231=19000. Ans. Proof, 138x138=19044 : ✓19044 138. 13. Suppose a lighthouse built on the top of a rock; the distance between the place of observation and that part of the rock level with the eye is 620 yards; the distance from the top of the rock to the place of observation is 846 yards; and from the top of the lighthouse, 900 yards; the height of the lighthouse is required. Ans.√900×900-620 × 620-√√/846×846—620×620= 76.77 yards. 14. Suppose one of those meteors called fire-balls, to move parallel to the earth's surface, and 50 miles from it, at the rate of 20 miles per second: in what time would it move round the earth? : Thus the earth's diameter is 7964 miles, then 7964+50 X 2 8064 the diameter of the circle described by the ball, then 8064 x 3.1416-25333.8624 miles its circumference, and 25333.8624 20 1266.69312 sec'ds=21 6 41 35 13 55 12 Ans. 15. The mean distances of the planets from the sun, in English miles, are as follows: Mercury 36686617.5, Venus 68552135.83, Earth 94772980 (or 95000000), Mars 144404783.33, Jupiter 492912533.33, Saturn 903957657.5; now, as a cannon-ball at its first discharge flies a mile in 8 seconds, and sound 1142 feet in a second, in what time, at the above rate, would a ball pass from the earth to the sun, and sound move from the sun to Saturn? Thus: 94772980 x 8"-758183840-24 years, 15 days, 6 hours, 27 minutes, 20 seconds, for the passage of the ball. And 903957657.5×5280=4772896431600 feet, and 4772896431600 1142 132 years, 192 days, 21 hours, 42 minutes, 21 seconds, for the passing of sound from the sun to Saturn. Ans. 16. Light passes from the sun to the earth in 8.2 minutes; in what time would it pass from the sun to the Georgium Sidus (Herschel), it being 1803930416.66 English miles? Ans. As 94772980: 8.2 :: 1803930416.66: 2 h. 36 m. 4" 50". 17. If a cubic foot of iron were hammered or drawn into a square bar, an inch about, that is, of an inch square, required its length, supposing there is no waste of metal. Thus, 12 x 12 × 12÷.25 × .25×4-6912 inches=576 ft. Ans. 18. How many square feet of boards in a load consisting of 150, 12 feet in length, and 11 inches in width? Thus 12.5x11=137÷12-11.46 feet in one board; 11.46 X150=1719 square feet (nearly), Ans. 19. The solid content of a square stone is found to be 136 feet, its length is 9 feet; what is the area of one end? and if the breadth be 3 feet 11 inches, what is the depth? Thus: 136.5 x 1728 9.5 × 12 area 2069.0526 in. and 2069.0526 47 44.022 inches. Ans. 20. How many perches of stone are there in a wall 48.25 feet in length, 1.5 in width, and 12 feet in height? Ans. 35.09+ perches. 21. Required the quantity of wood in a tier 64.5 feet in length, 4 feet in width, and 4 feet in height. Ans. 8.0625=81 cords. 22. In a pile of stone 25 feet in length, 12 feet in width, and 5 feet in height; required the cubical feet, the number of perches, and the value at 75c. per perch? Ans. 1718.75 cubical feet 69.444 perches, value D52.08.3+ 23. How much wood in a pile 14 feet in length, 61 feet high, and 34 feet in width; and value at D4 per cord? Ans. 2.47802 cords; D11.15.1090 value. 24. What is the breadth of a piece of cloth which is 36 yards long, and which contains 63 square yards? Ans. 63 36 13 yards. 25. How many yards of carpeting 14 yards wide will cover a floor 30 feet long and 22 feet broad? - Thus 30×22.5-675 ft.÷9=75 yds.÷11-60 yds. Ans. 26. What is the side of a square piece of land containing 289 square rods? 27. If the diameter of the earth be 7930 miles, what is the circumference? Ans. 24912.8. 28. How many miles does the earth move in revolving round the sun, supposing the orbit to be a circle whose diameter is 190 millions of miles? Ans. 596902654 29. What is the circumference of a circle whose diameter is 769843 rods? 30. If the circumference of the sun be 2800000 miles, what is the diameter ? Ans. 891267. 31. What is the diameter of a tree which is 5 feet round? 32. If the circumference of the moon be 6850 miles, what is her diameter ? Ans. 2180 miles. 33. If the whole extent of the orbit of Saturn be 5650 millons of miles, how far is he from the sun? 34. What is the area of a circle whose diameter is 623 feet: Ans. 304836 square feet. |