WRITTEN EXERCISES

531. 1. Extract the square root of 15625; of 1.5625.

After two figures of the root in the first process have been obtained, we find that we have subtracted, in all, 10000 + 4400, or 14400, the square of 12 tens, the part of the root already found. Therefore, regarding 12 tens as the first part of the root, the units of the root are obtained in the usual way.

When there are decimal figures, as in the second process, they are pointed off into periods of two figures each, beginning at the decimal point. The process is then the same as for integers.

Separate the number into periods of two figures each, beginning at units or at the decimal point.

Find the greatest square in the left-hand period, and write its root for the first figure of the required root.

Square this root, subtract the result from the left-hand period, and annex to the remainder the next period for a dividend.

Double the root already found, for a partial divisor, and by it divide the dividend, disregarding the right-hand figure. The quotient, or quotient diminished, will be the second figure of the

root.

Annex to the partial divisor for a complete divisor the figure last found, multiply this divisor by the figure of the root last found, subtract the product from the dividend, and to the remainder annex the next period for the next dividend.

Proceed in this manner until all the periods have been used. The result will be the square root sought.

1. When the number is not a perfect square, annex periods of decimal ciphers and continue the process.

2. The square root of a common fraction may be found by extracting the square root of numerator and denominator separately or by reducing it to a decimal and then extracting its root.

Extract the square root and express as a common fraction:

Verify the following:

21. √2=1.414+ 22. √3 = 1.732+

Extract the square root, to the nearest thousandth, of:

23. √5 = 2.236+

36. Find the side of a square whose area is 1 acre.

SOLUTION

Let x number of feet in one side of the square.

Then, the area = x2 square feet. But 1 acre = 43,560 square feet.
Therefore,

x2 = 43,560

Extracting the square root of both members,

x = 208.7+, the number of feet in the side.

Note that a square 209 feet on a side is only a little more than an acre.

Find, to within .1 ft, the side of a square whose area is :

37. 1.5 A.

38. 12,000 sq. ft.

THIRD PROG. AR.

20

39. 375,000 sq. ft.

40. A football field is 110 yd. long and 160 ft. wide. Find, to within .1 ft., the side of a square field having the same area. 41. Find the dimensions, in rods, of a 20-acre rectangular field whose length is twice its width.

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SUGGESTION. - Let x number of rods in width. Then 2 x number of rods in length, and x(2 x), or 2 x2, = number of square rods in area.

42. Find the dimensions, in rods, of a 90-acre rectangular field whose length is 4 times its width.

43. How much more fence is required to inclose the field mentioned in exercise 42 than a square field of equal area?

532. 1. Since the longest side, or hypotenuse, of this rightangled triangle is 5 units long, how many

square units are there in the square scribed upon the hypotenuse?

2. Since one of the other two sides, or legs, is 3 units long and the other 4 units long, how many square units are there in the square described upon each? in both these squares?

3. How does the number of units

de

3

5

4

in the square of the hypotenuse compare with the number of units in the sum of the squares of the other two sides?

The square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides.

WRITTEN EXERCISES

533. 1. Find the hypotenuse of this right-angled triangle.

SOLUTION.

x2 = 62 +82 = 36+ 64 = 100 ..x= 10

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2. Draw a right-angled triangle whose legs are 1 in. and 12 in. Compute the hypotenuse. Measure it. Compare results.

3. In the following right-angled triangles find the length

4. Draw an inch square and a straight line connecting two opposite corners, or a diagonal. Compute the length of the diagonal to the nearest .001 of an inch.

5. Draw a rectangle 4 in. by 71⁄2 in. Draw one of its diagonals and describe a square on it. Compute the area of the

square.

6. Draw a 5-inch square and one diagonal. How does the area of the square on the diagonal compare with the area of the 5-inch square?

7. Measure the length and width of a rectangular room. Measure a diagonal on the floor as accurately as you can. Then compute the length of the diagonal to the nearest .001 of a foot. Which is the more accurate method of finding the length of the diagonal when the length and width are known exactly?

8. Next measure the height of the ceiling. Compute the distance from the lower corner of the room at one end of the diagonal on the floor to the upper corner at the other end.

9. A 40-foot ladder leans against a wall, with the foot 6 feet from the base of the wall. Draw a sketch and compute the height of the top of the ladder.

10.. Two vessels sailed from the same point, one north at the rate of 15 knots an hour, the other east at the rate of 20 knots an hour. How far apart were they after 6 hours?

11. How far apart are the opposite corners of a square farm that contains 360 acres?

MENSURATION

534. The difference in direction of two lines that meet is called an angle (§ 241); the lines are called the sides, and the point where they meet, the vertex of the angle.

535. When a straight line meets another straight line forming two equal angles, each angle is called a right angle.

536. The two lines that form a right angle are said to be perpendicular to each other.

537. An angle that is less than a right angl is called an acute angle.

538. An angle that is greater than a right angle but less than two right angles is called an obtuse angle.

539. Lines that cannot meet, however far they are extended, are called parallel lines.

PLANE FIGURES

Two RIGHT Angles

ACUTE ANGLE

OBTUSE ANGLE

PARALLEL LINES

540. A surface such that a straight line joining any two points of it lies wholly in the surface is a plane surface.

541. A plane surface can be measured in only two directions, and hence has only two dimensions, length and breadth.

542. A portion of a plane surface bounded by four straight lines is called a quadrilateral.

You have learned how to measure and find the area (§ 244 - § 246) of squares and rectangles.