ART. 326. TO FIND THE AREA OF THE SURFACE OF A BODY BOUNDED BY PLANE SURFACES. Rule. Find the area of the surfaces separately; then add. TO FIND THE AREA OF THE CURVED SURFACE OF A RIGHT CONE. Rule.-Multiply the circumference of the base by the slant hight, and take half the product. TO FIND THE SURFACE OF A GLOBE. Rule.-Multiply the square of the diameter by 3.1416 1. Each side of the base of a triangular pyramid is 5 ft. 4 in.; its slant hight, from the vertex to center of each side of the base 7 ft. 6 in.: find the area of its surface. Ans. 60 sq.ft. 2. What is the convex surface of a cone, whose side is 25 ft., and diameter of the base 8 ft.? Ans. 333.795 sq. ft. 3. Find the area of the curved surface and base of a right cone, the slant hight 4 ft. 7 in.; the diameter of its base 2 ft. 11 in. Ans. 27.679895+sq. ft. 4. If the earth be a perfect sphere, 7912 mi. in diameter, what its superficial contents? Ans. 196663355.7504 sq. mi. ART. 327. GAUGING Is the method of finding the contents of any regular vessel, in gallons, bushels, barrels, &c. When the vessel is in the form of a cube or parallelopiped, apply this Rule. Take the dimensions in inches and multiply the length, breadth, and depth together; This product divided by 231, will give the contents in wine gal.; or, divided by 2150.4, will give the contents in bu. 1. How many wine gallons in a trough, 10 ft. long, 5 ft. wide, and 4ft. deep? Ans. 1496+gal. 2. How many bushels in a box, 12 ft. long, 6 ft. wide, and 10 ft. deep? Ans. 578.57+bu. ART. 328. TO FIND THE CONTENTS OF A CISTERN, BOTH ENDS Circular, the Upper and Lower Diameters EQUAL. Rule. Take all the dimensions in inches; then square the diameter, multiply this by the hight, and this product by .7854; this will give the contents in cubic inches, and this divided by 231 will give the contents in wine gallons, which may be reduced to barrels by dividing by 31.5. NOTE. Since .7854-231-.0034, therefore, when required to multiply by .7854 and divide the product by 231, shorten the operation (Art. 61), by multiplying at once by .0034. 1. How many barrels in a cistern, the diameter being 4 ft., the depth 6 ft. ? Ans. 17.9 bl. 2. How many barrels in a cistern, the diameter 6 ft., and depth 9 ft. ? Ans. 60.43+ bl. ART. 329. TO FIND THE CONTENTS OF A CISTERN, BOTH ENDS CIRCULAR, AND DIAMETERS UNEQUAL. Rule-Having the dimensions in inches, multiply together the diameters of the two ends; to the product add one-third of the square of the difference between these diameters: Then multiply this sum by the hight, and the product by .7854: the result will be cu. in., which reduce as in Art. 328. 1. What the contents, in wine gallons, of a cistern, the upper diameter 40 in.; lower diameter 30 in.; depth 50 in.? Ans. 209.66+ gal. 2. The contents in bl., of a cistern, the upper diameter 7 ft. 6 in.; lower diameter 10 ft.; depth, 12 ft. 6 in.? Ans. 1791⁄2 bl. ART. 330. TO FIND THE CONTENTS OF A CASK OR Barrel. When the staves are straight from the bung to each end, consider the cask as the frustums of two equal cones and find the contents by the rule, Art. 324. When the staves are curved, find the mean diameter of the cask, by adding to the head diameter, two-thirds of the difference between the bung and head diameters; or, If the staves are but little curved, add six-tenths of the difference; and, Having the mean diameter, find the solidity in the same manner as that of a cylinder, Art. 322. Since multiplying the square of half the mean diameter by 3.1416 is the same as to multiply its square by .7854; and multiplying by .7854 and dividing by 231, (Art. 328, Note,) is the same as to multiply by .0034; Therefore, to find the contents of a cask, when its dimensions are in inches, apply the following: Rule.-Multiply the square of the mean diameter by the length, and that product by .0034; the result will be wine gal. 1. How many gallons in a cask, the staves much curved, the bung diameter 40 in., head diameter 31 in., length 50 in. ? Ans. 232.73 gal. 2. Find the contents of a cask, the staves nearly straight, bung diameter 32 in., head diameter 30 in., length, 40 in. Ans. 132.38+ gal. ART. 330b. MISCELLANEOUS EXAMPLES. 1. A rectangular field is 15 rods long: what must be its breadth to contain one acre? Ans. 103 rd. 2. How many cubic feet in a room 24 ft. long, 18 ft. 6 in. wide, 10 ft. 7 in. high? Ans. 4699 cu. ft. 3. The area of a circle is 1 sq. ft.: what its diameter? Ans. 13.5405+ in. 4. The solid contents of a globe are 1 cu. ft. what the diameter ? Ans. 14.8884+ in. the area. 5. The sides of a triangle are 30, 40, and 50 feet: required Ans. 663 sq. yd. 6. How many sq. ft. in a plank 12 ft. 6 in. long; one end 15 in. broad, the other 11 in.? Ans. 1313 sq. ft. 7. Two circles, 10 and 16 ft. in diameter, have the same center: what their difference of area? Ans. 122.5224 sq. ft. 8. What will it cost to line a rectangular cistern, 6 ft. long, 2 ft. broad, 2 ft. 6 in. deep, with sheet lead, at 4 cts. a lb.; allowing 8 lb. of lead to each sq. ft. of surface? Ans. $16.64 9. At 25 cts. a bushel, what will oats cost to fill a bin 5 ft. long, 4 ft. wide, 4 ft. deep? Ans. $16.07+ 10. What is the area of a circle, of which the circumference is 1448 ft.? Ans. 3 A. 3 R. 12 P. 25 sq. yd. 8+ sq. ft. 11. Required the surface of a cube, each side being 37 in. Ans. 8214 sq. in. ART. 331. MECHANICAL POWERS.-This subject properly belongs to a more advanced work, and will be found appropriately treated in "Ray's Higher Arithmetic." ART. 332.-100 PROMISCUOUS QUESTIONS. 1. The sum of three equal numbers is 1236: what is one of the numbers? Ans. 412. 2. The sum of two equal numbers, less 225, is 675: find one of the numbers. Ans. 450. 3. There are four equal numbers, whose sum divided by 3, 292: find one of them. Ans. 219. 4. What cost 5 lb. 15 oz. of tea, at $1.20 per lb. ? Ans. $7.121 5. What cost 13 bu. 3 pk. potatoes, at $1.45 per bu.? Ans. $19.933 6. Two men, A and B, purchased a farm of 320 acres; A paid $1000, and B paid $600: how many acres should each receive? Ans. A, 200; B, 120 acres. 7. In what time will a man, walking at the rate of 33 miles an hour, travel 42 miles? Ans. 11 hr. 20 min. 8. What number multiplied by 13 will=143? A. 108. 9. I have a number in my mind, which 3, when 6: what is the number? = 81 less than Ans. 27. 10. A man bought 4yd. of cloth at $3 per yd., and 10 yd. at $3 per yd.: he paid with muslin at $per yd.: how many yards were required? Ans. 1111⁄2yd. 11. After spending 3 of my money and of what was left, I had $125 remaining: what sum had I at first? Ans. $500. 12. Multiply the sum of 27 and 1 by their difference, expressing the product decimally. Ans. 3.30078125 13. I was married at the age of 21: if I live 19 yr. longer, I will have been married 60 yr.: what is my age? Ans. 62 yr. 14. Find the least Com. Mult. of 8, 12, 21, 36, and 48, and divide it by the greatest Com. Div. of 65 and 143. Ans. 77-73. 15. How many French meters, each 39.371 English inches, are there in 3 mi. 5 fur. 110 yd.? Ans. 5934.317+ 16. In what time can you count 800000000, at the rate of 250 a min., counting 10 hr. a da., 365 da. to the yr.? Ans. 14 yr. 223 da. 3hr. 20 min. Ans. .7575. 17. Divide 12.625 by 163. 18. What does the rent of a house amount to from May 20, 1854, to May 10, 1855, at $250 per year? Ans. $2431. 19. I bought an equal quantity of flour, butter, and sugar, for $47; the sugar was 12 cts., the butter 30 cts., and the flour 5 cts., a pound: how much of each did I buy? Ans. 100 lb. 20. A cistern is full of water; after 35 gal. are taken out, it is full: how many gal. will it contain? Ans. 120 gal. 21. I bought 60 barrels of flour at $5 a barrel; sold 23 barrels at $4 a bl.: at how much per bl. must I sell the rest, to gain $51 on the whole? Ans. $7. 22. How many boxes of 3 qr. 13 lb. each, can be filled from a hhd. of sugar containing 12 cwt. 1 qr. 7 lb. ? Ans. 14. 23. What will it cost to gild a globe 10 inches in diameter, at 5 cents per square inch? Ans. $15.708 24. If 1 ox is worth 8 sheep, and 3 oxen are worth 2 horses, what is the value of each horse, the sheep being valued at $2.50 each? Ans. $30. 25. If of $1 buy of an ox, what will 26. What number has to 54 the same ratio that 19 has to 9? of a sheep, and of a sheep be worth Ans. 114. 27. Two-thirds of the ratio of to 3, is three times the ratio of 3 to what? Ans. 1. 28. By working 13 hr. a day, a man can perform a piece of work in 5 days: in what time can he perform it by working 9 hr. a day? Ans. 77 da. 29. I bought 50lb. of tea for $40, and sold it so as to clear $15: had I purchased $100 worth of tea, and sold it at the same rate, what sum would I have made? Ans. $37.50 30. A clock gains 71 min. in 24 hr. on Monday: what will be the time by following Thursday evening? It is set right at noon it at 6 o'clock on the Ans. 6 hr. 243 min. 31. If 7 men can mow 35 acres of grain in 4 days, how many acres will 10 men mow in 3 days? Ans. 433 A. 32. Bought 35 yd. linen at $3 per yd., and sold 161yd. at $1 per yd., and the rest at $ per yd.: what the gain by the transaction? Ans. $5.67+ 33. If a man can build 10 cu. ft. of wall in an hr., what length of wall, 5 ft. high, and 2 ft. thick, can he build in 6 da., working 11 hr. a da. ? Ans. 66 ft. |