64 in. To complete the cube, it also requires three oblong rectangular blocks, whose length is 60 inches, and whose end is 4 inches square (Fig. 4); also a cube, whose edge is 4 inches (Fig. 6). The side of one of the oblong blocks being 60 X 4, one side of the three will be 3 times 60 × 4=720 square inches, and one side of the small cube will be 42 16 square inches. If, now, we multiply the sum of these surfaces, 10800+720+16, 11536 (Fig. 7), by their thickness, 4, and increase the cube 216000 6. What is the cube root of 122097755681? 10. What is the cube root of .000729? 11. What is the cube root of 329778750? 12. What is the cube root of .57? 125? Ans. 4961. Ans. 9002. Ans. 5.8. Ans. 690.8+. Ans. .8291+. Ans. 3.185+. Ans. 1.587+. Ans. z. 17. What is the cube root of? (A1 = 15) 18. What is the cube root of 40? 19. What is the cube root of 1}? OPTIONAL EXAMPLES. Ans. Ans. 3. Ans. .1957+ Ans. 1.04004. NOTE. In the following, the pupil need extract the root to but four places, if decimal fractions be reached. 40. 5.43 X 194 +27054036008=? 41. Find the difference between the sum of the cube roots of 13824 and .000729, and the cube root of their sum. 408. PRACTICAL EXAMPLES. 1. What is the length of one side of a cubical block of granite which contains 7077888 solid inches? 2. What will be the edge of a cubical pile of wood, composed of 1000 loads, each 8 feet long, 4 wide, and 24 feet high? 3. What will be the length of a cubical pile of wood that will contain one cord? 4. What will be the length of a cube which will contain as much as another whose edge is 15 feet? Ans. 7.5 feet. 5. What is the depth of a cubical cistern which will contain 9 times as much as one whose depth is 5 feet? Ans. 10.4004+ feet. 6. What must be the dimensions of a cubical vessel that shall contain 300 gallons of water, reckoning 231 cubic inches to a gallon? Ans. 41.075+. 7* What will be the cost of boards, at $11.25 per thousand feet, to construct the bottom and sides of a cubical bin which shall contain 75 bushels of grain? NOTE.-2150.4 cubic inches1 bushel. Ans. $1.191+. 8. What will be the cost of lead, at $.124 per lb., there being 1 lbs. to the square foot, to line a cubical box containing 15§ cubic feet? 9. How many yards of paper, 1 yard wide, will be required to line 98 cubical boxes, each containing 5 cubic feet? Ans. 384 yards. 10* The walls of the ancient city of Babylon are said to have been 350 feet high, and built of brick; the city, 15 miles square inside the walls. Suppose the average thickness of the walls to have been 60 feet, what would be the length of a cubical pile composed of the brick in the walls? For Dictation Exercises, see Key. Ans. 1881.2+ feet MENSURATION. 409. The definitions of various surfaces and solids are found on pages 109, 112, 114. Such as are in general use, and not there found, are given in this section. PLANE SURFACES, RECTILINEAR FIGURES. TRIANGLES. 410. The Right-angled Triangle contains one right angle. 411. The Obtuse-angled Triangle contains one obtuse angle. M M N Right-angled. NL Obtuse-angled. 412. The Equilateral Triangle contains three equal sides 413. The Isosceles Triangle contains two equal sides. 414. The Scalene Triangle has no sides equal. 415. A Parallelogram is a quadrilateral whose opposite sides are parallel. 416. A Rhombus is a parallelogram whose sides are all equal, and whose angles are not right angles. 417. A Trapezoid is a quadrilateral only two of whose sides are parallel. 418. A Rectangle is a parallelogram whose angles are right angles. 419. A Square is a rectangle whose sides are all equal. 420. A Trapezium is a quadrilateral of which no two sides are parallel. Rectangle. Square. Trapezium. 421. The term Polygon is a general name applied to any figure bounded by straight lines. 422. The Base of a figure is the line upon which it is supposed to stand. 423. The Altitude of a figure is its height. The lines M N in the preceding figures indicate altitudes. Polygon. 424. The Diagonal of a figure is a line joining any two angles not adjacent. The lines O P are diagonals. AREAS. 425. The area of a square or rectangle equals the product of its length and its breadth or height. (Art. 173.) 426. The area of any parallelogram equals the product of its base and its height; for it can be proved to be equal to a rectangle of the same base and height. 427. The area of a triangle equals half B the product of its base and height; for every triangle equals one half of a parallelogram of the same base and height. |