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355. A Cone is a solid, having a circle for its base, and tapering uniformly to a point, called the vertex.
The Altitude of a pyramid and of a cone is a line drawn from the vertex perpendicular to the plane of the base; as B C.
The Slant Hight of a cone is a line drawn from the A vertex to the circumference of the base; as A C.
356. To find the convex surface of a pyramid or of a
RULE. - Multiply the perimeter or the circumference of the base by half its slant hight, and the product is the convex surface.
1. How many yards of cloth, that is 27 inches wide, will it require to cover the sides of a pyramid whose slant hight is 100 feet, and whose perimeter at the base is 54 feet ?
Ans. 400 yards. 2. Required the convex surface of a cone, whose slant hight is 50 feet, and the circumference at its base 12 feet.
Ans. 300 square feet. 357. To find the solidity or volume of a pyramid or of a cone.
RULE. Multiply the area of the base by one third of its altitude, and the product will be the solidity.
1. The largest of the Egyptian pyramids is square at its base, and measures 693 feet on a side. Its altitude is 500 feet. Now, supposing it to come to a point at its vertex, what are its solid contents, and how many miles in length of wall would it make, 4 feet in hight and 2 feet thick ?
Ans. 80041500 cubic feet ; 1894.9 miles in length. 2. What are the solid contents of a cone, whose hight is 30 feet, and the diameter of its base 5 feet ? Ans. 196.3+ feet.
358. A Frustum of a Pyramid is the part that remains after cutting off the top, by a plane parallel to the base.
355. What is a cone? What is the altitude of a pyramid and of a cone ? The slant hight of a cone? — 356. The rule for finding the surface of a pyramid and of a cone ? — 357. The rule for finding the solidity of a pyramid and of a cone ? - 358. What is the frustum of a pyramid ?
359. A Frustum of a Cone is the part that remains after cutting off the top, by a plane parallel to the base.
360. To find the surface of a frustum of a pyramid or of a cone.
RULE. Add the perimeters or the circumferences of the two ends together, and multiply this sum by half the slant hight. Then add the areas of the two ends to this product, and their sum will be the surface.
1. There is a square pyramid, whose top is broken off 20 feet slant hight from the base. The length of each side at the baze is 8 feet, and at the top 4 feet; what is its whole surface ?
feet. 2. There is a frustum of a cone, whose slant hight is 12 feet, the circumference of the base 18 feet, and that of the upper end 9 feet; what is its whole surface?
Ans. 194.22+ square feet. 361. To find the solidity or volume of a frustum of a pyramid or of a cone.
RULE. · Find the area of the two ends of the frustum ; multiply these two areas together, and extract the square root of the product. To this root add the two areas, and multiply their sum by one third of the altitude of the frustum; the product will be the solidity.
1. What is the solidity of the frustum of a square pyramil, whose hight is 30 feet, and whose side at the bottom is 20 feet, and at the top 10 feet ?
Ans. 7000 cubic feet. 2. What are the contents of a stick of timber 20 feet long, and the diameter at the large end being 12 inches, and at the smaller end 6 inches ?
Ans. 9.162+ feet.
THE SPHERE. 362. A Sphere is a solid, bounded by one continued convex surface, every part of which is equally distant from a point within, called the center.
The Axis or Diameter of a sphere is a line passing through the center, and terminated by the surface.
359. What is the frustum of a cone ? - 360. The rule for finding the sur: face of a frustum of a pyramid or of a cone? - 361. The rule for finding the solidity of a frustum of a pyramid or of a cone ? - 362. What is a sphere? The diameter or axis of a sphere ?
363. To find the surface of a sphere.
RULE. – Multiply the diameter by the circumference, and the product will be the surface.
1. What is the convex surface of a globe, whose diameter is 20 inches ?
Ans. 1256.6+ square
inches. 2. If the diameter of the earth is 8000 miles, what is its convex surface ?
Ans. 201061888 square miles. 364. To find the solidity or volume of a sphere.
RULE. Multiply the surface by 1 of the diameter, or multiply the cube of the diameter by .523598, and the product will be the solidity. NotĖ. — .523598 is 4 of 3.141592.
1. What is the solidity of a sphere whose diameter is 20 inches ?
Ans. 4188.7+ inches. 2. If the diameter of a globe or sphere is 5 feet, how many cubic feet does it contain ?
Ans. 65.44+ cubic feet. 365. To find how large a cube may be cut from any given sphere, or be inscribed in it.
RULE. Square the diameter of the sphere, divide the product by 3, and extract the square root of the quotient for the answer.
1. How large a cube may be inscribed in a sphere 10 inches in diameter?
Ans. 5.773+ inches. 2. What is the side of a cube that may be cut from a sphere 30 inches in diameter ?
Ans. 17.32+ feet.
THE SPHEROID. 366. A Spheroid is a solid, generated by the revolution of an ellipse about one of its diameters.
If the ellipse revolves about its longer or transverse diameter, the spheroid is prolate, or oblong; if about its shorter or conjugate diameter, the spheroid is oblate, or flattened.
367. To find the solidity or volume of a spheroid.
RULE. Multiply the square of the shorter axis by the longer axis, and this product by.523598, if the spheroid is prolate. Or,
363. The rule for finding the surface of a sphere? 364. The rule for find. ing the solidity of a sphere ? – 365. The rule for finding how large a cube can be cut from a given sphere ? — 366. What is a spheroid ? A prolate spheroid? An oblate spheroid ? - 367. The rule for finding the solidity of a spheroid ?
If it is oblate, multiply the square of the longer axis by the shorter axis, and this product by .523598.
1. What is the solidity of a prolate spheroid, whose transverse axis is 30 feet, and the conjugate axis 20 feet?
Ans. 6283.17+ cubic feet. 2. What is the solidity of an oblate spheroid, whose axes are 30 and 10 feet ?
Ans. 4712.38+ cubic feet.
MENSURATION OF LUMBER.
368. Boards are usually measured by the square foot. Planks, joists, beams, &c., are usually surveyed by board measure, the board being considered to be 1 inch in thickness.
Round timber is sometimes measured by the ton, and sometimes by board measure.
369. To find the contents of a board.
RULE. — Multiply the length of the board, taken in feet, by its width, taken in inches, and divide this product by 12; the quotient is the contents in square feet.
Note. - If the board is tapering, take half the sum of the width of its ends for the width.
1. What are the contents of a board 18 inches wide and 16 feet long?
Ans. 24 feet. 2. What are the contents of a board 24 feet long and 30 inches wide ?
Ans. 60 feet. 370. To find the contents of joists, beams, &c.
RULE. Multiply the width, taken in inches, by the thickness, and this product by the length, in feet ; divide the last product by 12, and the quotient is the contents in feet.
1. What are the contents of a joist 4 inches wide, 3 inches thick, and 12 feet long?
Ans. 12 feet. 2. What are the contents of a square stick of timber 25 feet long and 10 inches thick ?
Ans. 2081 feet. 371. To find the contents of round timber.
RULE. Multiply the length of the stick, taken in feet, by the square of one fourth the girt, taken in inches ; divide this product by 144, and the quotient is the contents in cubic feet.
368. By what measure are planks, joists, &c., usually surveyed ? What is the usual thickness of a board ? How is round timber ineasured ? - 369. The rule for finding the contents of a board ? — 370. The rule for finding the contents of joists, &c.
Note 1. — The girt of tapering timber is usually taken about one third the distance from the larger to the smaller end.
Note 2 · A ton of timber, estimated by this method, contains 50,16% cubic feet.
1. How many cubic feet of timber in a stick, whose length is 50 feet, and whose girt is 60 inches ? Ans. 78} cubic feet.
2. What are the contents of a stick, whose length is 30 feet, and girt 30 inches ?
Ans. 11.7+ solid feet.
MISCELLANEOUS QUESTIONS. 1. What number is that, to which if } be added, the sum will be 71 ?
Ans. 73. 2. What number is that, from which if 37 be taken, the remainder will be 44 ?
Ans. 733 3. What number is that, to which if 3 be added, and the sum divided by 5%, the quotient will be 5 ?
Ans. 239. 4. From II of a mile takes of a furlong.
Ans. 4fur. 12rd. 8ft. Sin. 5. John Swift can travel 7 miles in g of an hour, but Thomas Slow can travel only 5 miles in 11 of an hour. Both started from Danvers at the same time for Boston, the distance being 12 miles. How much sooner will Swift arrive in Boston than Slow ?
Ans. 1239 seconds. 6. If & of a ton cost $ 49, what will lcwt. cost ?
Ans. $ 3.92. 7. How many bricks 8 inches long, 4 inches wide, and 2 inches thick, will it take to build a wall 40 feet long, 20 feet high, and 2 feet thick ?
Ans. 43200 bricks. 8. How many bricks will it take to build the walls of a house, which is 80 feet long, 40 feet wide, and 25 feet high, the wall to be 12 inches thick ; the brick being of the same dimensions as in the last question ?
Ans. 159300 bricks. 9. How many tiles, 8 inches square, will cover a floor 18 feet long, and 12 feet wide ?
Ans. 486 tiles. 10. If it cost $ 18.25 to carry 11cwt. 3qr. 191b. 46 miles, how much must be paid for carrying 83cwt. 2qr. 11lb. 96 miles ?
Ans. $ 266.702 374. 11. A merchant sold a piece of cloth for $ 24, and thereby lost 25 per cent. ; what would he have gained had he sold it for $ 34?
Ans. 67 per cent.