781. To find the three dimensions of a rectangular solid, the volume and the ratio of the dimensions being given. 1. What are the dimensions of a rectangular solid, whose volume is 4480 cu. ft., and its dimensions are to each other as 2, 5, and 7 ? OPERATION.–4480(2 x 5 x 7) = 4 ft. ; 4x2 = 8 ft., height; 4x5= 20 ft., width; 4x 7 = 28 ft., length. RULE.-1. Divide the volume by the product of the terms proportional to the three dimensions, and extract the cube root of the quotient. II. Multiply the root thus obtained by cach proportional term ; the products will be the corresponding sides. 2. What are the dimensions of a rectangular box whose volume is 3000 cu. ft., and its dimensions are to each other as 2, 3, and 4 ? 3. A pile of bricks in the form of a parallelopiped contains 30720 cu. ft., and the length, breadth, and height are to each other as 3, 4, and 5. What are the dimensions of the pile ? 4. Separate 405 into three factors, which shall be to each other as 2, 23, and 3. Ans. 6, 73, and 9. 782. Similar Solids are such as have the same form, and differ from each other only in volume. PRINCIPLES.—1. The volumes of similar solids are to each other as the cubes of their like dimensions. If the volume of a ball 3 in. in diameter is 27 cu. in., what is the volume of a ball 7 in. in diameter? OPERATION.—33 : 3 :: 27 cu. in. : x = 343 cu. in., volume. 2. The like dimensions of similar solids are to each other as the cube roots of their volumes. If the diameter of a ball whose volume is 27 cu. in. is 3 in., what is the diameter of a ball whose volume is 343 cu. in.? OPERATION.—27: 343 :: 38 : 2x3 = 373; 3/373 = 7 in., diameter. ft., 1. What is the edge of a cube whose entire surface is 1050 sq. and what is its volume ? 2. What must be the inner edge of a cubical bin to hold 1250 bushels of wheat ? Ans. 11 ft. 7 in. 3. How many globes 4 in. in diameter are equal to one whose diameter is 12 inches ? 4. How many gallons will a cistern hold, whose depth is 7 ft., the bottom being a circle 7 ft. in diameter and the top 5 feet in diameter ? Ans. 1494.25 gal. 5. What is the value of a stick of timber 24 ft. long, the larger end being 15 in. square, and the less 6 in., at 28 cents a cubic foot ? 6. If a cubic foot of iron were formed into a bar 1 an inch square, without waste, what would be its length ? Ans. 576 ft. 7. How many barrels of 314 gal, will a cistern hold that is 8.3 ft. in diameter, and 7 feet deep? 8. If a log 18 ft. long and 3 ft. in diameter is hewn square, how many cubic feet does it contain? 9. Find the volume of a cube, the area of whose entire surface is 7 sq. inches. Ans. 1 cu. ft. 469 cu. in. 10. If a marble column 10 in. in diameter contains 27 cu. ft., what is the diameter of a column of equal length that contains 81 cubic feet ? Ans. 14.42 in, 11. Supposing the earth to be a perfect sphere 7912 miles in diameter, what is its volume in cubic miles ? 12. How many board feet in a post 11 ft. long, 9 in. square at the bottom, and 4 in. square at the top ? Ans. 40 ft. 7. 13. The surface of a sphere is the same as that of a cube, the edge of which is 12 in. Find the volume of each, 14. The contents of a cubical block of marble are 4913 cu. ft. Find the superficial contents or surface. Ans. 1014 sq. ft. 15. Find the dimensions of a bin that holds 450 bu. of grain, if the width and depth are equal, and the length 3 times the width. 16. A ball 4.5 in. in diameter weighs 18 oz. Avoir. ; what is the weight of a ball of the same density, that is 9 in. in diameter ? ft. 6 sq. 17. In what time will a pipe supplying 6 gal. of water a minute, fill a tank in the form of a hemisphere, that is 10 ft. in diameter? 18. If the altitude of a cone that weighs 640 lb. is 8 ft., what is the altitude of a similar cone that weighs 270 lb. ? 19. If a stack of hay 8 ft. high weighs 8 cwt., what is the weight of a similar stack that is 24 ft. high? • Ans. 216 cwt. 20. The diameter of a cistern is 8 ft.; what must be its depth to contain 75 hhd. of water ? Ans. 12.56 ft. 21. If a cable 3 inches in circumference supports a weight of 2500 pounds, what must be the circumference of a cable that will support 4960 pounds ? Ans. 3.77 in., nearly. 22. How many bushels in a heap of grain in the form of a cone, whose base is 8 ft. in diameter and altitude 4 feet? GAUGING. 784. Gauging is the process of finding the capacity or volume of casks and other vessels. For ordinary purposes the diagonal rod is used, which gives only approximate results. A cask is equivalent to a cylinder having the same length and a diameter equal to the mean diameter of the cask. 785. To find the mean diameter of a cask (nearly). RULE. Add to the head diameter š, or, if the staves are but little curved, .6, of the difference between the head and bung diameters. 786. To find the volume of a cask in gallons. RULE. Multiply the square of the mean diameter by the length (both in inches) and this product by .0034. 1. How many gallons in a cask whose head diameter is 24 in., bung diameter 30 in., and its length 34 inches? OPERATION.—24 + (30 — 24 x ) = 28 in., mean diameter. 282 x 31x.0034= 90.63 gal., capacity. 2. What is the volume of a cask whose length is 40 in., the diameters 21 and 30 in., respectively? Ans. 99.14 gal. 3. How many gallons in a cask of slight curvature, 3 ft. 6 in. long, the head diameter being 26 in., the bung diameter 31 in. ? CIRCLES. =the circumference. 787. 3.1416, the product by Divided , .8862, the product by Multiplied by * .3183, the quotient , the quotient = the side of an equal sq're. = the side of an inscribed by { , the quotient } = equilateral triangle = the side of an inscribed square. 2. The radius of any circle Multiplied 6.28318 = the circumference. Divided .15915 } Multiplied } by { 1.2732 , the quotient Multiplied } by { 3.1416 , the quotient 3.1416 , the quotient } = the diameter. 3. The square of the diameter of any circle = the area. .3183, the product Divided „ .2821, the product = the side of an equal sq're. Divided 5 3.5450, the quotient Multiplied S.2756, the product = the side of the inscribed } by Divided 73.6276, the quotient equilateral triangle. Multiplied 1 5 .2251, the product = the side of an inscribed Divided s 4.4428, the quotients square. 5. The square of the circumference of any circle Multiplied .07958, the product = the area. Divided 12.5663, the quotient 6. The area of any circle Multiplied 5 1.2732, the product = the square of the diam. Divided S , The square of the radius of any circle x 3.1416 7. Half the circumference of a circle x 4 its diameter Square of the circumference of a circle x.07958 } by {1:2732, the product } = area. 789. The Unit of land measure is the acre. Measurements of land are commonly recorded in square miles, acres, and hundredths of an acre. PROBLEMS. 1. What is the value of a farm 189.5 rods long and 150 rods wide, at $42an acre ? 2. A man having a field 70 rd. square appropriated 5 A. of it to corn, 100 sq. rd. to garden vegetables, and the remainder to meadow. What fractional part of the whole field did the meadow comprise ? 3. I bought a piece of land 16 ch. long and 15 ch. wide, at $100 an acre, and dividing it into lots of 6 rods by 5 rods, sold them at $50 each. What was my gain? Ans. $4000. 4. At $2.75 a rod, how much less will it cost to fence a piece of land 80 rods square, than if the same were in the form of a rectangle twice as long and one-half as wide ? Ans. $220. |