854 24. How many cubic feet in a wall 100 feet long, 15 feet high, and 3 feet thick? 25. A man wishes to make a cubical bin, which shall contain 19683 solid feet: what must be the length of its side? 26. If a stick of timber containing 400 cubic feet, is 60 feet long, and 3 feet thick, what is its width? 439. A prism is a solid whose bases or ends are similar, equal, and parallel, and whose sides are parallelograms. OBS. 1. When the bases of a prism are triangles it is called a triangular prism; when square, a square prism, &c. 2. All prisms whose ends are parallelograms are called parallelopipedons. 3. A right prism is one whose sides are perpendicular to its bases. All other prisms are called oblique. 440. To find the solidity of a prism. Multiply the area of the base by the height. (Leg. VII. 12.) OBS. 1. The height of a prism is the perpendicular distance between the planes of the bases. Hence, in a right prism, the height is equal to the length of one of the sides. 2. This rule is applicable to all prisms, triangular, quadrangular, &c.; also to all parallelopipedons, 27. What is the solidity of a prism whose base is 5 feet square, and its height Ans. 375 cubic feet. 15 feet? 28. What is the solidity of a triangular prism whose height is 20 feet, and the area of whose base is 460 square feet? 441. To find the lateral surface of a right prism. OBS. If we add the areas of both ends to the lateral surface, the sum will be the whole surface of the prism. 29. Required the lateral surface of a triangular prism whose perimeter is 4 Ans. 54 square inches. inches, and its length 12 inches. 30. Required the lateral surface of a quadrangular prism whose sides are each 2 feet, and its length 19 feet. 442. A pyramid is a solid whose base is a triangular, square, or polygonal plane, and whose sides terminate in a point called the vertex. (Fig. 5.) A frustum of a pyramid is the part which remains after cutting off the top by a plane parallel to the base, as a b c d e, Fig. 5. Fig. 5. S E C Fig. 6. 443. A cone is a solid whose base is a circle, and its sides terminate in a point called the vertex. (Fig. 6.) A frustum of a cone is the part which remains after cutting off the top by a plane parallel to the base, as a b c d. 444. To find the solidity of a pyramid, or cone. (Leg. VII. 18. VIII. 4.) Multiply the area of the base by } of the altitude. 31. Required the solidity of a square pyramid, the A side of whose base is 25 feet, and whose height is a b D B 32. Required the solidity of a cone, the diameter of whose base is 30 feet, and whose height is 90 feet. 445. To find the lateral or convex surface of a regular pyramid, or cone. (Leg. VII. 16. VIII. 3.) Multiply the perimeter of the base by the slant-height. OBS. The slant-height of a regular pyramid, is the distance from the vertex or summit to the middle of one of the sides of the base. 33. What is the lateral surface of a triangular pyramid whose s'ant-height is 10 feet, and each side 8 feet? Ans. 120 feet. 34. What is the convex surface of a cone, the perimeter of whose base is 500 yards, and slunt-height 120 yards? 446. To find the solidity of a frustum of a pyramid, or cone. (Leg. VII. 19. Sch. VIII. 6.) To the sum of the areas of the two ends, add the square root of the product of these areas; then multiply this sum by of the perpendicular height. 35. The areas of the ends of a frustum of a cone are 9 square feet, and 4 square feet, its height 15 feet: what is its solidity? Ans. 95 feet. 36. The two ends of a frustum of a pyramid aro 4 feet and 3 feet square, its height 10 feet: what is its solidity? 447. The convex surface of a frustum of a pyramid, or cone, is found by multiplying half the sum of the circumferences of the two ends by the slant-height. (Leg. VII. 17.) 37. The circumferences of the two ends of a frustum of a pyramid are 12 feet and 8 ft., and its slant-height 7 ft.: what is its convex surface? Ans. 70 sq. ft. 448. A cylinder is a long circular body of uniform diameter, whose ends are equal parallel circles. (Fig. 7.) Fig. 7. 449. To find the solidity of a cylinder. (Leg. VIII. 2.) Multiply the area of the base by the height or length. 39. Required the solidity of a cylinder 6 feet in diameter, and 20 feet high. Ans. 565.488 cubic feet. 40. Required the solidity of a cylinder 30 feet in diameter, and 65 feet long. 450. To find the convex surface of a cylinder. Multiply the circumference of the base by the height. 41. What is the convex surface of a cylinder 16 inches in circumference and 40 inches long? Ans. 640 square in. 42. What is the convex surface of a cylinder, whose diameter is 20 feet, aud its height 65 feet? 451. A sphere or globe is a solid terminated by a curve surface, every part of which is equally distant from a certain point within called the centre. 452. To find the surface of a sphere or globe. Multiply the circumference by the diameter. (Leg. VIII. 9.) 3. Required the surface of a globe 13 inches in diameter. Ans. 531 sq. in. nearly. Required the surface of the earth, its diameter being 8000 miles. 453. To find the solidity of a sphere or globe. Multiply the surface by of the diameter. Ans. 904.77792 in. What is the solidity of a globe 12 in. in diameter ? 46. What is the solidity of the earth, its diameter being 8000 miles. GAUGING OF CASKS. 454. To find the contents or capacity of casks. Multiply the square of the mean diameter into the length in inches; then this product multiplied into .0034 will be the wine gallons, or multiplied into .0028 will be the beer gallons. OBS. The mean diameter of a cask is found by adding to the head diameter .7 of the difference between the head and bung diameters when the staves are very much curved; and by adding .5 when very little curved, and by adding .65 when they are of a medium curve. 47 How many wine gallons does a cask contain whose length is 35 inches, its bung diameter 30 inches, and its head diameter 26 inches, it being but little curve!? Ans. 93.296 gallons. 48. How many beer gallons in a cask 54 inches long, whose bung diameter is 42 inches, and head diameter 35 inches, its staves being much curved? MISCELLANEOUS EXAMPLES. } 1. A farmer having sold and of his sheep, had 95 left: how many had he at first? 2. A man having $15750, spent for a house, the remainder for a barn, and of the balance for a carriage: how much had he left ? 3. What is the difference between 18 of 275, and 3 of 315. 4 What number is that, of which exceeds by 387? 8. What must be added to 217 that the sum may be 17 times 19}} 9. What number multiplied by 45}, will produce 288‡? 10. What number divided by 374, will give 1931 for the quotient! 11. Bought of a ship, and sold of it: how much was left? 12. A broker negotiated a bill of exchange of $10360, at 18 per cent. what was his commission? 13. At what per cent. must $6376 be loaned, to draw $135.49 interest in 3 months? 14. What sum must be loaned at 7 per cent., to gain $455 interest in 3 months? 15. What sum must be loaned at 6 per cent. interest, to gain $650 semi-annually? 16. In what time will $8284 gain $365, at 6 per cent. interest? 17. At 7 per cent. int., in what time will $857.25 double itself? 18. At what per cent. interest will $500 double itself in 10 years? 19. What is the present worth of $1365, payable in 6 months, when money is worth 7 per cent. per annum? 20. At 6 per cent. discount, what is the present worth of $1623.28, due in 1 year? 21. What is the bank discount on a note of $730, payable in 4 months, at 6 per cent.? 22. What is the bank discount on a note of $1575, payable in 60 days, at 7 per cent.? 23. What will 35 shares of railroad stock cost, at 10 per cent. advance? Ans. $3867.50. 24. What cost 63 shares of bank stock, at 3 per cent. discount} 25. What premium must a man pay annually for insuring $8500 on his store and goods, at 1 per cent.? 26. If I obtain insurance on goods, worth $16265, at 2 per cent., and the goods are lost, how much shall I lose i 27. Paid $78.75 insurance annually, which was 1 per cent. on the sum insured: what was the amount of my policy? 28. Paid $24.54 insurance on $6544: what was the per cent. ? 29. Received $862.50 dividend on $17250 stock: what was the per cent.? 30. Bought a farm for $5640, spent $258 upon it, then sold it for 15 per cent. profit: how much did it sell for? 31. Bought goods for $4390, and sold them on 6 mos. at 22 per cent. above cost: what was the profit, allowing 7 per cent. Interest? 82. Bought 15000 gals. oil for $8500: allowing 1 per cent. leakage, how must it be sold to gain 15 per cent. 33. If I buy 1675 yards of flannel for $368.50, how must I retail it per yard to gain 25 per cent.? Ans. 27 cts. 34. A grocer bought 2500 lbs. of coffee for $250, and sold it at 6 per cent. loss: what did he get per pound? 35. A merchant bought 1824 yds. of cloth, at $2.50 per yard and retailed it at $3 per yard: what per cent. was his profit, and how much did he make? 36. A shop-keeper bought 100 pieces of lace, for $250, and sold them for $375: what per cent. did he make, and how much? 37. If a grocer buys 3680 lbs. of cheese, at 4 cts. per. lb., and Bells it at 6 cts., what per cent. and how much is his profit? 38. What is the ad valorem duty, at 333 per cent., on a quantity of cloths which cost $10436? 39. What is the ad valorem duty, at 15 per cent., on a cargo of tea, invoiced at $35856 ? 40. At 37 per cent., what is the duty on a quantity of silks which cost $23265? 41. The sum of two numbers is 856, and their difference is 75: what are the numbers? 42. The sum of two numbers is 5643, and their difference is 125: what are the numbers? 43. The difference of two numbers is 63, and the smaller number is 365 what is the greater number? 44. The product of two numbers is 3750, and one of the numbers s 75: what is the other? 45. What number is that, 46. What number is that, of which is 34 times 45 ? of of which is 63 times 3 of 120? 47. How long will it take a person to count a billion, if he counts 50 a minute, and works 6 hours per day, for 5 days a week, and 52 weeks a year i |