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3. What is the square root of 278784 ?

278784(528 Answer.

25 102)287

204 1048) 8384

8384 4. What is the square root of 776161 ? Ans. 881. 5. What is the square root of 998001 ? Ans. 999. 6. What is the square root of 10342656 ? Ans. 3216. 7. What is the square root of 9645192360241 ?

Ans. 3105671. 8. Extract the square root of 234.09. Ans. 15.3. 9. Extract the square root of .000729. Ans. .027. 10. Required the square root of 17.3056. Ans. 4.16. 11. Required the square root of 373. Ans. 19.3132079+. 12. Required the square root of 8.93. Ans. 2.98831055+. 13. What is the square root of 1.96 ?

Ans. 1.4. 14. Extract the square root of 3.15. Ans. 1.77482393+. 15. What is the square root of 572199960721 ?

Ans. 756439. If it be required to extract the square root of a vulgar fraction, reduce the fraction to its lowest terms; then extract the square root of the numerator for a new numerator, and of the denominator for a new denominator; or reduce the vulgar fraction to a decimal, and extract its root. 16. What is the square root of ?

Ans. 1 17. What is the square root of 7%?

Ans. TT 18. What is the square root of 424?

Ans. 64. 19. What is the square root of 5216

Ans. 71 20. What is the square root of 9517 ? 21. What is the square root of 36336ı

? 22. Extract the square root of 6.

Ans. 2.5298+ 23. Extract the square root of 84. Ans. 2.9519+ 24. Required the square root of 2. Ans. 1.41421+:

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Ans. 9. Ans. 1915

APPLICATION OF THE SQUARE ROOT.

DEFINITIONS.

1. A circle is a figure bounded by a line equally distant from a point called the centre.

2. A triangle is a figure of three sides. 3. An equilateral triangle is that whose three sides are equal.

4. An isosceles triangle is that which has two sides equal, and a perpendicular from the angle where the equal sides meet bisects the base.

5. A right-angled triangle is a figure of three sides and three angles, one of which is a right angle. See figure 1.

Fig. 1.

The longest side is called the hypothenuse, the horizontal side the base, and the other side is called the perpendicular.

Hypothenuse.

Perpendicular.

Base.

Fig. 2.

The following propositions are demonstrably true. In a right-angled triangle, the square of the longest side is equal to the sum of the squares of the other two sides.

In all similar triangles, that is, in all trian. gles whose corresponding angles are equal, the sides about the equal angles are in direct proportion to each other; that is, as the longest side of one triangle is to the longest side of the other, so is either of the other sides of the former triangle to the corresponding side of the latter. Let ABC and D E F be

Fig. 3.

Fig. 4. two similar triangles, and let A B be 6 feet, B C, 8 feet, and A C, 10 feet. Again, let D E be 12 feet, EF, 16 feet, and D F will

20

16 be 20 feet. That is, BC will be to EF as A B to

10 DE. Then 8 feet : 16

اتر

D

E

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12

feet :: 6 feet : 12 feet. Again, A B will be D E as A C to DF. That is, 6 feet : 12 feet :: 10 feet : 20 feet.

Circles are to each other as the squares of their diameters.

If the diameter of a circle be multiplied by 3.14159, the product is the circumference.

If the square of the diameter of a circle be multiplied by .785398, the product is the area.

If the square root of half the square of the diameter of a circle be extracted, it is the side of an inscribed square.

If the area of a circle be divided by .785398, the quotient is the square

of the diameter.

EXAMPLES.

square ?

25. A certain general has an army of 141376 men. How many must he place in rank and file to form them into a

Ans. 376. 26. If the area of a circle be 1760 yards, how many

feet must the side of a square measure to contain that quantity ?

Ans. 125.857+ feet. 27. If the diameter of a round stick of timber be 24 inches, how large a square stick may be hewn from it?

Ans. 16.97+ inches. 28. I wish to set out an orchard of 2400 mulberry-trees, so that the length shall be to the breadth as 3 to 2, and the distance of each tree, one from the other, 7 yards. How many trees must there be in the length, and how many in the breadth ; and on how many square yards of ground will they stand ?

Ans. 60 in length; 40 in breadth ; 112749 square yards. 29. If a lead pipe of an inch in diameter will fill a cistern in 3 hours, what should be its diameter to fill it in 2 hours ?

Ans. .918+ inches. 30. If a pipe 14 inches in diameter will fill a cistern in 50 minutes, how long would it require a pipe that is 2 inches in diameter to fill the same cistern?

Ans. 28m. 71sec. 31. If a pipe 6 inches in diameter will draw off a certain quantity of water in 4 hours, in what time would it take 3 pipes of four inches in diameter to draw off twice the quantity ?

Ans. 6 hours. 32. If a line 144 feet long will reach from the top of a fort to the opposite side of a river that is 64 feet wide, what is the height of the fort ?

Ans. 128.99+ 33. A certain room is 20 feet long, 16 feet wide, and 12 feet high ; how long must a line be to extend from one of the lower corners to an opposite upper corner ? Ans. 28.28 feet.

34. Two ships sail from the same port; one goes due north 128 miles, the other due east 72 miles ; how far are the ships from each other?

Ans. 146.86+. 35. There are two columns in the ruins of Persepolis left standing upright; one is 70 feet above the plane, and the other 50; in a straight line between these stands a small statue, 5 feet in height, the head of which is 100 feet from the summit of the higher, and 80 feet from the top of the lower column. Required the distance between the tops of the two columns.

Ans. 143.537+ feet. 36. The height of a tree, growing in the centre of a circular island, 100 feet in diameter, is 160 feet; and a line extending from the top of it to the farther shore is 400 feet. What is the breadth of the stream, provided the land on each side of the water be level ?

Ans. 316.6+ feet. 37. A ladder 70 feet long is so planted as to reach a window 40 feet from the ground, on one side of the street, and without moving it at the foot it will reach a window 30 feet high on the other side ; what is the breadth of the street ?

Ans. 120.69+ feet. 38. If an iron wire to inch in diameter will sustain a weight of 450 pounds, what weight might be sustained by a wire an inch in diameter ?

Ans. 45,000lbs. 39. A tree 80 feet in height stands on a horizontal plane ; at what height from the ground must it be cut off, so that the top of it may fall on a point 40 feet from the bottom of the tree, the end where it was cut off resting on the stump?

Ans. 30 feet. 40. Four men, A, B, C, D, bought a grindstone, the diameter of which was 4 feet; they agreed that A should grind off his share first, and that each man should have it alternately until he had worn off his share ; how much will each man grind off ?

Ans. A 3.22+,,B 3.81+, C 4.97+, D 12 inches. 41. What is the length of a rope that must be tied to a horse's neck, that he may feed over an acre ? Ans. 7.136+ rods.

42. Required the greatest possible number of hills of corn that can be planted on a square acre, the hills to occupy only a mathematical point, and no two hills to be within three and a half feet of each other.

Ans. 4165. 43. James Page has a circular garden, 10 rods in diameter ; how

many trees can be set upon it, so that no two shall be within ten feet of each other, and no tree within two and a half feet of the fence inclosing the garden ?

Ans. 241.

44. I have a board whose surface contains 49 square feet; the board is 14 inches thick, and I wish to make a cubical box of it. Required the length of one of its equal sides.

Ans. 36 inches.* 45. A carpenter has a plank 1 foot wide, 2231 feet long, and 21 inches thick; and he wishes to make a box whose width shall be twice its height, and whose length shall be twice its width. Required the contents of the box.

Ans. 5719 cubic inches. *

* The operation may be found on page 203 of the last edition of the Key.

SECTION LXII.

EXTRACTION OF THE CUBE ROOT.

A CUBE is a square prism, being bounded by six equal sides, which are perpendicular to each other.

A number is said to be cubed when it is multiplied into its square.

To extract the cube root is to find a number, which, multiplied into its square, will produce the given number.

Rule. - 1. Separate the given number into periods of three figures each, by putting a point over the unit figure, and every third figure beyond the place of units.

2. Find by the table the greatest cube in the left-hand period, and put its root in the quotient.

3. Subtract the cube thus found from this period, and to the remainder bring down the next period; call this the dividend,

4. Multiply the square of the quotient by 300, calling it the triple square; multiply also the quotient by 30, calling it the triple quotient ; the sum of these call the divisor.

5. Find how many times the divisor is contained in the dividend, and place the result in the quotient.

Note. — One or two units, and sometimes three, must be allowed.

6. Multiply the triple square by the last quotient figure, and write the product under the dividend ; multiply the square of the last quotient fig. ure by the triple quotient, and place this product under the last ; under all, set the cube of the last quotient figure, and call their sum the subtrahend.

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