7x7×0.7854 1.069 square feet for area of end. 3. How many cubic inches in a round bar of iron, 20 feet long and 4 of an inch in diameter? Ans. 106 029 cubic inches. PROBLEM VIII.—To find the volume of a pyramid, or of a cone. RULE. Multiply the area of the base by one-third the altitude. (Geometry, B. VII., Prop. XVII.; and B. VIII. Prop. V.) EXAMPLES. 1. The Egyptian pyramid, Cheops, covers a square of 763% feet on a side, and is 480 feet perpendicular height. How many cubic feet does it contain? Ans. 932447294 cubic feet. PROBLEM XI. To find the volume of a frustum of a pyramid, or of a cone. RULE. Find a mean proportional between the area of the two bases, to which add the sum of the bases, and multiply the result by one-third the altitude of the frustum. EXAMPLES. 1. Suppose a cistern in the form of a frustum of a months, and the remaining on a credit of 9 months. How much ready money ought to pay the purchase, interest being 7 per cent. ? Ans. $1159.64, nearly. 2. In the above example, by the principles of equation of payments, how much credit ought I to have on the whole sum of $1200? Ans. 6 months. 3. Now, what is the present worth of $1200 due at the end of 6 months, interest being 7 per cent.? Ans. $1159-42, nearly. 4. I employed A and B to ditch my meadow. A was to receive 87 cents per rod, and B was to have 1121 cents per rod; each wrought until his wages amounted to $50. What was the amount of ditch dug by both? Ans. 10133. 5. Three merchants, A, B, and C, enter into partnership A advances $1200, B $800, and C $600. A leaves his money 8 months, B 10 months, and C 14 months in the business. They gain $500. What is the share of each? A receives $184. of his;, 6. A and B have the same income; A saves but B, by spending $120 per annum more than A, at the end of 10 years finds himself $200 in debt. the income? What was Ans. $500. 7. Suppose a book to contain 365 pages, averaging 40 lines of 10 words each on each page. How many words would the book contain? Ans. 146000 words. 8. There are 31173 verses in the Bible; how many days will it require to read it through, if 30 verses are read daily? Ans. 1039 days. 9. After expending of my money, and of the remainder, I had remaining $72; how much had I at first? Ans. $128. 10. If I sell cloth at $1.50 per yard, and gain 25 per cent., how ought I to have sold it so as to lose 20 per cent.? Ans. $0.96. cent. 11. Sold cloth at $1.50 per yard, and gained 25 per What should I have lost per cent., if I had sold it at $0.96 per yard? Ans. 20 per cent. 12. If I buy cloth at $1.20 per yard, how must I sell it so as to gain 25 per cent.? Ans. $1.50. 13. A merchant has to make the following payments at three different periods: $2832 in 3 months, $2560 in 9 months, and $1450 in 16 months. The creditor wishes to receive the whole sum of $6842 at once. When ought the payment to be made? Ans. In 8 months. 14. A father gives to his five sons $1000, which they are to divide according to their ages, so that each elder son shall receive $20 more than his next younger brother. What is the share of the youngest? Ans. $160. 15. A company of 90 persons consists of men, women, and children. The men are 4 in number more than the women, the children 10 more than the adults. How many men, women, and children, are there in the company? 16. The common school fund for the state of New-York was $1975093.15 in 1843, and during the same year there were in the state, 677995 children between the ages of 5 and 16 years. How much would the above fund amount to per scholar? Ans. $291, nearly. 17. The whole number of volumes in the common school libraries of New York, in 1843, was 874865. What would be their value at 37 cents per volume? Ans. $328074.37. 18. The whole number of children taught in N. Y. during the year 1843, was 657782, and the whole number of schools was 10860. How many scholars on an average would each school consist of? Ans. Between 60 and 61. 19. Suppose the Erie canal to be 60 feet wide, and 6 feet deep; how many miles in length will it require to make one cubic mile of water? Ans. 77440 miles. of a copper mine, sells of his What, at this rate, is the value Ans. $4000. 20. A person owning interest in it for $1800. of the whole? 21. Suppose I buy a certain lot of oranges at 3 cents apiece, and as many more at 5 cents apiece, and sell them at 4 cents apiece; do I gain or lose by the operation? Ans. I neither gain nor lose. 22. Suppose I buy a certain number of oranges at 3 for one cent, and as many more at 5 for one cent, and sell them at 4 for one cent; do I gain or lose by the operation? I lose of a cent on each orange. If the whole number of oranges was 60, I should lose one cent. Ans. { 23. Suppose I expend a certain sum of money for oranges at of a cent. apiece and another equal sum for another lot of oranges at of a cent apiece, and sell them at of act. apiece, do I gain or lose by the operation? Ans. I neither gain nor lose. 24. Suppose I expend a certain sum of money for oranges at 3 cents apiece, and another equal sum for another lot at 5 cents apiece; how much do I gain on each cent expended, if I sell them at 4 cents apiece? Ans. I gain of a cent on each cent employed in the purchase. If the whole sum employed was 15 cents, I should gain 1 cent. 25. If A can do a piece of work in 3 days, B in 4 days, and C in 5 days, how many times longer will it take B to do it alone, than it will take A and C together to do it? Ans. 2 times. 26. If A can accomplish a piece of work in of a day, Bin of a day, and C in of a day, how many times longer will it take B to do it alone, than it will take A and C together to do it? Ans. 2 times. 27. What is the shortest piece of cloth which shall be at the same time, an even number of yards, an even number of Ells Flemish, an even number of Ells English, and an even number of Ells French? Ans. 60 quarters=15 yards. 28. A man died, leaving $1000, to be divided between his two sons, one 14, and the other 18 years of age, in such a proportion, that the share of each being put to interest at 6 per cent., should amount to the same sum when they should arrive at the age of 21. What did each one receive? Since the shares of each would amount to equal sums when they should come of age, it is obvious that they must have been to each other reciprocally as the amount of $1 for the respective times 7 years and 3 years. amount of $1 for 7 years at 6 per cent., is $1.42. amount of $1 for 3 years at 6 per cent., is $1.18. Hence, The The |