18. If an ox, that weighs 800lbs., girts 6 feet, what is the weight of an ox that girts 7 feet ? Ans. 1270.3lbs. 19. If a tree, that is one foot in diameter, make one cord, how many cords are there in a similar tree, whose diameter is two feet? Ans. 8 cords. 20. If a bell, 30 inches high, weighs 1000lbs., what is the weight of a bell 40 inches high ? Ans. 2370.3lbs. 21. If an apple, 6 inches in circumference, weighs 16 ounces, what is the weight of an apple 12 inches in circumference ? Ans. 128 ounces. Section 50. GEOMETRICAL PROBLEMS. 1. To find the area of a square or parallelogram. Rule. Multiply the length by the breadth, and the product is the superficial contents. 2. To find the area of a rhombus or rhomboid. Rule. Multiply the length of the base by the perpendicular height. 3. To find the area of a triangle. Rule. Multiply the base by half the perpendicular height ; or, add the three sides together ; then take half of that sum, and out of it subtract each side severally ; multiply the half of the sum and these remainders together, and the square root of this product will be the area of the triangle. 4. Having the diameter of a circle given, to find the circumference. Rule. Multiply the diameter by 3.141592, and the product is the circumference. Note. The exact proportion, which the diameter of a circle bears to the circumference, has never been discovered, although some mathematicians, have carried it to 200 places of decimals. If the diameter of a circle be 1 inch, the circumference will be 3.141592653 5897932384626433832795028941971693993751058209749445923078164062 8620899862803482534211706798214808651328230664709384464609550518 22317253594081284802 inches nearly. 5. Having the diameter of a circle given, to find the side of an equal square. Rule. Multiply the diameter by.886227, and the product is the side of an equal square. 6. Having the diameter of a circle given, to find the side of an equilateral triangle inscribed. Rule. Multiply the diameter by :707016, and the product is the side of a triangle inscribed. 17. Having the diameter of a circle given, to find the area. Rule. Multiply the square of the diameter by .785398, and the product is the area. Or, multiply half the diameter by half the circumference, and the product is the area. 8. Having the circumference of a circle given, to find the diameter. Rule. Multiply the circumference by .31831, and the product is the diameter. 9. Having the circumference of a circle given, to find the side of an equal square. Rule. Multiply the circumference by .282094, and the product is the side of an equal square. 10. Having the circumference of a circle given, to find the side of an equilateral triangle inscribed. Rule. Multiply the circumference by .2756646, and the product is the side of an equilateral triangle inscribed. 11. Having the circumference of a circle given, to find the side of an inscribed square. Rule. Multiply the circumference by .225079, and the product is the side of a square inscribed. 12. To find the contents of a cube or parallelopipedon. Rule. Multiply the length, height, and breadth, continually together, and the product is the contents. 13. To find the solidity of a prism. RULE. Multiply the area of the base, or end, by the height. 14. To find the solidity of a cone or pyramid. RULE. Multiply the area of the base by of its height. 15. To find the surface of a cone. Rule. Multiply the circumference of the base by half its slant height. 16. To find the solidity of the frustum of a cone, or pyramid. Rule. Multiply the diameters of the two bases together, and to the product add 1 of the square of the difference of the diameters; then multiply this sum by .785398, and the product will be the mean area between the two bases ; lastly, multiply the mean area by the length of the frustum, and the product will be the solid contents. Or, find when it would terminate in a cone, and then find the contents of the part supposed to be added, and take it away from the whole. 17. To find the solidity of a sphere or globe. RULE. Multiply the cube of the diameter by .5236. 18. To find the convex surface of a sphere or globe. Rule. Multiply its diameter by its circumference. 19. To find the contents of a spherical segment. RULE. From three times the diameter of the sphere, take double the height of the segment ; then multiply the remainder by the square of the height, and the product by the decimal .5236 for the contents ; or to three times the square of the radius of the segment's base, add the square of its N* height; then multiply the sum by the height, and the product by .5236 for the contents. 20. To find how large a cube may be cut from any given sphere, or be inscribed in it. Rule. Square the diameter of the sphere, divide that product by 3, and extract the square root of the quotient for the answer. 21. To find the number of gallons, &c., in a square vessel. Rule. Take the dimensions in inches; then multiply the length, breadth, and height together ; divide the product by 202 for ale gallons, 231 for wine gallons, and 2150.42 for bushels. 22. To find the contents of a cask. Rule. Take the dimensions of the cask in inches ; viz. the diameter of the bung and head, and the length of the cask. Note the difference between the bung diameter and the head diameter. If the staves of the cask be much curved between the bung and the head, multiply the difference by .7; if not quite so much curved, by .65; if they bulge yet less, by .6; and, if they are almost straight, by .55; add the product to the head diameter ; the sum will be a mean diameter by which the cask is reduced to a cylinder. Square the mean diameter thus found, then multiply it by the length ; divide the product by 359 for ale or beer gallons, and by 294 for wine gallons. 23. To find the contents of a round vessel, wider at one end than the other. Rule. Multiply the greater diameter by the less ; to this product, add of the square of their difference, then multiply by the height, and divide as in the last rule. 24. To measure round timber. Rule. Multiply the length of the stick, taken in feet, by the square of the girt, taken in inches ; divide this product by 144, and the quotient is the contents in cubic feet. Note. The girt is usually taken about the distance from the larger to the smaller end. 1. What are the contents of a board 25 feet long and 3 feet wide ? Ans. 75 feet. 2. What is the difference between the contents of two floors ; one is 37 feet long and 27 feet wide, and the other is 40 feet long and 20 feet wide ? Ans. 199 feet. 3. The base of a rhombus is 15 feet, and its perpendicular height is 12 feet ; what are its contents ? Ans. 180 feet. 4. What are the contents of a triangle, whose base is 24 feet, and whose perpendicular height is 18 feet? Ans. 216 feet. 5. What are the contents of a triangular piece of land, whose sides are 50 rods, 60 rods, and 70 rods ? Ans. 1469.69+ rods. 6. What is the circumference of a circle, whose diameter is 50 feet ? Ans. 157.0796+ feet. 7. We have a round field 40 rods in diameter ; what is the side of a square field, that will contain the same quantity ? Ans. 35.44+ rods. 8. What is the side of an equilateral triangle, that may be inscribed in a circle 50 feet in diameter ? Ans. 35.35+ feet. 9. If the diameter of a circle be 200 feet, what is the Ans. 31415.92+ feet. 10. What is the diameter of a circle, whose circumference is 80 miles ? Ans. 25.46+ miles. 11. I have a circular field 100 rods in circumference ; what must be the side of a square field, that shall contain the same area ? Ans. 28.2+ rods. 12. Required the side of a triangle, that may be inscribed in a circle, whose circumference is 1000 feet. Ans. 275.66+ feet, 13. How large a square field may be inscribed in a circle, whose circumference is 100 rods ? Ans. 22.5+ rods square. 14. How many cubic feet are there in a cube whose sides are 8 feet ? Ans. 512 feet. 15. What is the difference between the number of cubic feet in a room 30 feet long, 20 feet wide, and 10 feet area ? |