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553. To find the surface of a sphere.
1. Find the surface of a sphere whose diameter is 9 in. OPERATION. -9 in. x 3.1416 28.2744 in., circumference. 28.2744 in. x 9 = 254.4696 sq. in., surface.
RULE. Multiply the diameter by the circumference of a great circle of the sphere.
2. What is the surface of a globe 16 in. in diameter ?
3. Find the area of the surface of a sphere whose circumference is 31.416 ft. Ans. 314. 16 sq. ft.
554. To find the volume of a sphere.
1. Find the volume of a sphere whose diameter is 18 in. OPERATION. — 18 in. × 3.1416 = 56.5488 in., circumference. 56.5488 in. x 18 1017.8784 sq. in., surface. 1017.8784 sq. in. x 18÷6 = 3053.6352 cu. in., volume.
RULE. Multiply the surface by of the diameter, or of the radius.
1. The Surface
2. Find the volume of a globe whose diameter is 30 in.
3. Find the volume of a globe whose circumference is 31.416 ft.
SUMMARY OF SPHERES.
2. The Volume
3. The Diameter
4. The Circumference
5. The Radius
1. One half the sum of two numbers is 800, and one half the difference of the same numbers is 200. What are the numbers? Ans. 1000 and 600. 2. What number is that to which, if you add of 3 of itself, the sum will be 61 ? Ans. 55.
3. What part of a day is 3 h. 21 min. 15 sec.?
Ans. 161 1152. 4. A commission merchant received 70 bags of wheat, each containing 3 bu. 3 pk. 3 qt. How many bushels Idid he receive?
5. Four men, A, B, C, and D, are in possession of $1100; A has a certain sum, B has twice as much as A, C has $300, and D has $200 more than C. How many dollars has A? Ans. $100.
6. At a certain election, 3000 votes were cast for three candidates, A, B, and C; B had 200 more votes than A, and C had 800 more than B. How many votes were cast for A?
7. What part of 17 is 31?
8. A merchant bought a hogshead of liquor for $28.35. How much water must be added to reduce the first cost to 35 cents per gallon? Ans. 18 gal.
9. A and B traded with equal sums of money; A gained a sum equal to of his stock; B lost $200, and then he had as much as A. How much was the original
stock of each ?
10. The difference between & and 7 of a number is 10. What is the number? Ans. 560.
11. A farmer sold 17 bu. of barley, and 13 bu. of wheat for $31.55; he received for the wheat 35 cents a bushel more than for the barley. What was the price of each per bushel? Ans. Barley, $.90; wheat, $1.25. 12. What is the interval of time between March 20, 21 minutes past 3 o'clock P.M., and April 11, 5 minutes past 7 o'clock A.M.? Ans. 21 da. 15 h. 44 min. 13. What o'clock is it when the time from noon is of the time to midnight? Ans. 5 o'clock 24 min. P.M. 14. What is the least number of gallons that can be shipped in either hogsheads, tierces, or barrels, just filling the vessels, without deficit or excess? Ans. 126 gal.
15. A ferryman has four boats: one will carry 8 barrels, another 9, another 15, and another 16. What is the smallest number of barrels that will make full freight for any one and all of the boats?
16. A and B have the same income; A saves of his, but B, by spending $30 a year more than A, at the end of four years finds himself $40 in debt. What is their income, and how much does each spend a year?
Ans. Inc. $160. A spends $140. B, $170. 17. If a load of plaster weighing 1825 lb. costs $2.19, how much is that per ton of 2000 lb.? Ans. $2.40.
18. If 2 yards of cloth 13 yd. wide cost $3.37%, what will be the cost of 361⁄2 yd. 11⁄2 yd. wide? Ans. $52.779. 19. I lend my neighbor $200 for 6 months. How long ought he to lend me $1000 to balance the favor?
20. I bought railroad stock to the amount of $2356.80, and found that the sum invested was 40% of what I had ieft. What sum had I at first? Ans. $8248.80. 21. 20% of of a number is what per cent of & of it? Ans. 12%.
22. Divide a prize of $10200 among 60 privates, 6 subaltern officers, 3 lieutenants, and a commander, giving to each subaltern double the share of a private, to each lieutenant 3 times as much as to the subaltern, and to the commander double that of a lieutenant. How much is each man's share? Ans. Com. $1200; each man, $100. 23. A is 51 miles in advance of B, who is in pursuit of him; A travels 16 miles per hour, and B 19 miles. In how many hours will B overtake A?
24. A mixes 3 gal. water with 12 gal. vinegar, at 50 cts. a gal. What is 1 gal. of the mixture worth? Ans. 40 cts. 25. If 240 bushels of wheat are purchased at the rate of 18 bushels for $22, and sold at the rate of 22 bushels for $33.75, what is the profit on the whole?
26. My horse, wagon, and harness together are worth $169; the wagon is worth 4 times the harness, and the horse is worth double the wagon. What is the value of each? Ans. Horse, $104. Wagon, $52. Harness, $13.
27. The shadow of a tree measures 42 ft.; a staff 40 in. in length casts a shadow 18 in. at the same time. What is the height of the tree? Ans. 931 ft.
28. If a piece of land 40 rd. long and 4 rd. wide makes an acre, how wide must it be to contain the same if it is but 25 rd. long? Ans. 63 rd.
29. A, B, and C are employed to do a piece of work for $26.45; A and B together are supposed to do 8 of it; A and C, and B and C. If they are paid proportionally, how much must each receive?
Ans. A, $11.901; B, $5.29; C, $9.253. 30. A certain principal, at compound interest for 5 years, at 6%, will amount to $669.113. In what time will the same principal amount to the same sum at 6% simple interest? Ans. 5 yr. 7 mo. 19.3+ da.
31. If 12 oz. of wool make 2 yd. of cloth that is 6 quarters of a yard wide, how many pounds of wool will it take for 150 yd. of cloth 1 yd. wide?
32. Six persons, A, B, C, D, E, and F, are to share among them $6300; A is to have 4 of it, B, C 3, D is to have as much as A and C together, and the remainder is to be divided between E and F in the proportion of 3 to 5. How much does each one receive?
33. What is the amount of $200 for 8 yr. at 6% compound interest? Ans. $318.769.
34. A garrison, consisting of 360 men, was provisioned for 6 months; but at the end of 5 months they dismissed so many of the men that the remaining provision lasted 5 months longer. How many men were sent away?
35. I paid $148.352 for 9728 ft. of pine lumber. How much was that per thousand?
36. Comparing two numbers, 483 was found to be their least common multiple, and 23 their greatest common divisor. What is the product of the numbers compared? Ans. 11109.
37. Eight workmen, laboring 7 h. a day for 15 da., were able to execute of a job; in how many days can they complete the rest, by working 9 h. a day, if 4 workmen are added to their number? Ans. 155 da.
38. A, B, and C traded in company; A put in $1 as often as B put in $ 3, and B put in $2 as often as C put in $5; B's money was in twice as long as C's, and A's twice as long as B's; they gained $52.50. How much was each man's share of the gain?
Ans. A's, $12. B's, $18. C's, $22.50. 39. If a hall 36 ft. long and 9 ft. wide requires 36 yd. of carpeting 1 yd. wide to cover the floor, how many yards 14 yd. wide will cover a floor 60 ft. long and 27 ft. wide? Ans. 144 yd.