SPHERE AND LARGEST CUBE. h' is obviously both diameter of the sphere and hypotenuse of the erect right triangle, h', h, p; h = = : h2 is the hypotenuse of the horizontal tri- = = Taking the square root, we have b = V10. Hence the formula: = = When the diameter 1, the side of the cube or .3333, etc..57735+, and the formula becomes: Side of cube = diameter x .57735. PROBLEMS. 1. What is the volume of a pyramid whose base is a rectangle 13 by 14 feet, and whose height is 18 feet? 2. What is the volume of a cylinder 108 in. in diameter and 10 ft. long? 3. What is the lateral surface of a cone whose base is 10 ft. in diameter and slant height 20 ft.? Find also the entire surface. 4. Find the surface of a sphere whose radius is 12 inches. 5. How many gallons will a hollow globe contain whose inside diameter is 20 inches? 6. What is the lateral surface of a triangular prism whose sides are each 6 feet and whose altitude is 8 feet? 7. What is the lateral surface of a quadrangular pyramid whose base is 15 feet square and the slant height 18 feet? 8. What is the lateral surface of a cone whose base is 10 ft. in diameter and whose slant height is 10 ft.? 9. Find the volumes in problems No. 6, 7, and 8. 10. Required the surface of the frustum of a cone whose slant height is 12 feet, diameter of lower base 10 ft. and upper base 6 feet. What is the volume? 11. Find the entire surface of the frustum of a triangular pyramid whose slant height is 40 in., and the sides of the upper base 4 in. and the lower base 10 in. 12. Required the contents of a cannon ball whose diameter is 9 inches. What is the surface? 13. At 45 cents a square foot, how much will it cost to gild a ball 25 inches in diameter ? 14. Find how many cubic inches of iron there are in a hollow sphere, the diameter being 15 inches long and the shell 3 inches thick? 15. A cylindrical can is 6 inches deep and 4 inches in diameter. If a cone of the same height and diameter be placed in the can, how much water will be required to fill the remaining space? 16. In the above problem, what is the ratio of the volume of the cone and cylinder? Does this show why 3 is used in the formula for the volume of a cone? 17. Find the side of the greatest square that can be inscribed in a circle whose diameter is 10 feet? 18. Find the edge of the greatest cube that can be cut from a wooden ball whose diameter is 5.5 inches. 19. I have a cubical box whose faces each contain 64 square inches. Find the diameter of the sphere that will exactly contain the box. 20. I have a circular garden whose circumference is 31.416 rods. I wish to reduce it within the circumference to the largest possible square form. Find the area of the square. GENERAL REVIEW. The following problems have been selected from the examination papers of the University of the State of New York. They are introduced here for the purpose of affording a complete review of the principles and methods set forth in the previous pages of the book. It is suggested that the best efforts of both teacher and pupil be applied to these problems, and that the science and art of arithmetic, as already illustrated, be faithfully recalled, studied afresh, and securely fixed in mind. Let every solution, therefore, proceed systematically, and every principle involved be distinctly stated. 1. Define sum, and illustrate your definition by a practical example. 2. A man deposits in bank $986.46. At different times he has drawn the following amounts: $314.18, $49.25, $57.62, $39.84, $25.13. Find the amount remaining in the bank. 3. Find the least number of bushels of grain that can be exactly measured either by a 3-quart, a peck, a 20-quart, or a bushel measure. 4. Reduce 29265 to its lowest terms. 22967 5. Simplify X 1.25 5 4.25 and express the result both as a common and as a decimal fraction. 6. Define composite number and give an example. 7. Make a receipted bill for the following: Harold Kirby bought of Pliny Hall, 10 lb. sugar at 5 cts., lb. tea at 60 cts., 3 lb. coffee at 40 cts., 1 sack flour at $1.50. 8. If the shadow of a post 6 ft. high is 4 ft. 6 in. long, what is the height of a tree whose shadow at the same time is 125 ft. long? (Solve by analysis.) 9. What would it cost to dig a cellar 80 ft. × 35 ft. × 8 ft. at $.84 per cubic yard? 10. A railway train runs Find its velocity per hour? of a mile in of a minute. (Solve by analysis.) 11. Define quotient, and give an illustration. 12. Find the prime factors of 1001 and 1309, and from these factors form the G. C. D., and the L. C. Dd. (least common multiple) of the two numbers. 13. A field 10 chains 50 links long and 8 chains 40 links wide produces 40 bushels of oats per acre; what is the value of the crop at 35 cents a bushel? 14. Find the sum of 93, 81, 5%, and 5. Express the result both as a fraction in lowest terms and as a decimal. 15. What part of an ounce (apothecaries' weight) is 5 drachms and 2 scruples? 16. Find the cost of a stick of timber 40 ft. long, 12 in. wide, 9 in. thick, at $12.50 per M., board measure. 17. A roll of wall paper 8 yd. long and 18 in. wide costs 25 cts. What will be the cost of paper for the four walls of a room 30 ft. × 27 ft. × 9 ft., no allowance being made for openings? 18. I bought 240 barrels of apples at $1.75 a barrel; lost 40 barrels through frost; at what price a barrel must I sell the remainder to gain 25% on the money invested? 19. If 2 men plough 15 acres in 5 days, working 10 hours a day, how many acres will 3 men plough in 4 days, working 8 hours a day? 20. Define greatest common divisor and least common dividend (multiple). Illustrate. 23. What part of a bushel is contained in a rectangular box 3 in. deep and 4 in. square? [A bushel cu. in.] 2150.42 24. From sixty subtract forty-seven and sixteen ten-millionths and express the decimal as a common fraction. 25. Find the cost of carpeting a room 18 ft. long, 15 ft. wide, with carpet 27 in. wide, at 75 cts. a yard. 26. Define divisor, root, proportion, fraction. 27. I retail oranges at 3 cts. each, gaining 150% on the purchase price. What did the oranges cost a dozen? 28. I sell an article at an advance of 25% on the cost and then discount the bill 5% for cash payment. My net gain is $63.75. Find the cost. 29. A cubic foot of water weighs 623 lb. Find the weight of a barrel of water. 30. On a bill of goods amounting to $485.50 I receive commercial discounts of 15%, 10%, and 5%. Find the net cost of the goods. 31. What principal loaned for 1 yr. and 3 mo. at 6% simple interest will amount to $1000? 32. A 30-day note discounted at a New York bank yields $358.02. What was the face of the note? 33. A note for $500 at 90 days, with interest at 6%, is discounted at a bank 30 days after it is dated. Find the proceeds. 34. A certain stock pays annual dividends of 4%. At what rate must it be bought to pay 5% on the investment? 35. Find the square root of 4,004,231 to two places of decimals. |