6. If light travels 186,000 mi. per second, and it takes it 8.158 sec. to come from the sun to the earth, what is the distance traveled?

7. Water is composed of two gases, oxygen and hydrogen, 88.89% by weight being oxygen. What is the weight of the hydrogen in a cubic foot of water?

8. The air is composed of two gases, oxygen and nitrogen. In every cubic foot of air there are 345.6 cu. in. of oxygen. What per cent of the volume of the air is nitrogen?

9. A certain cirrus cloud is observed to be 6 mi. above the earth, or 240% higher than a certain rain cloud observed a few hours before. How high was the rain cloud?

10. At the temperature when sound travels 1120 ft. per second, what is the distance of a thunder cloud in which lightning is seen 17 sec. before the thunder is heard?

11. If a large drop of rain falls at the rate of 20 ft. per second, and a small one, blown by the wind, only 25% as fast, how long will it take the small one to reach the earth from a cloud 24 mi. high?

12. The wind pressure in a hurricane has been known to be as great as 49.2 lb. per square foot. In such a storm how many tons pressure on the side of a large office building 104 ft. long and 308 ft. high?

13. In a great storm the velocity of the wind often reaches 88 ft. per second. What is then its velocity per minute? In a hurricane it has been known to be 66%% greater. What is then its velocity per hour?

14. When the mercury in the barometer is at 30 in., the pressure of air on every square inch of surface is 15 lb. What is the pressure on a pane of glass 2 ft. by 3 ft., when the increased air pressure forces the mercury up to 31 in.? Why does the glass not break?

POWERS AND ROOTS

560. Square numbers and square roots. If a square has a side 4 units, it has an area 16 square units. Therefore 16 is called the square of 4, and 4 the square root

of 16.

561. Square roots of areas. sidering the abstract numbers

sides and area,

Therefore, conrepresenting the

The side of a square is the square root of its area.

562. Writing squares and roots. The square of 4 is written 42; the square root of 16 is written √16.

563. Perfect squares. A number like 16 is a perfect square, but 10 is not a perfect square. We speak, however, of √103.16+, because 3.162 nearly equals 10.

564. Square roots of perfect squares. Square roots of perfect squares may often be found by factoring.

3)441

3)147

7)49

7

ORAL EXERCISE

State the square roots of the numbers in Exs. 1–8:

What are the sides of squares whose areas are as follows?

State the perimeters of squares whose areas are:

15. 1.21 sq. ft. 16. 0.49 sq. ft.

17. 169 sq. in.

WRITTEN EXERCISE

By factoring, find the square roots of the following:

Find the sides of squares whose areas are as follows:

Find the perimeters of squares whose areas are as follows:

28. 16,384 sq. in.

29. 16,900 sq. ft. 30. 129,600 sq. in.

31. From the corner of a square piece of land containing 576 sq. rd. a small square lot containing 64 sq. rd. is cut out. Draw the plan of the lots and find the perimeter

of each.

32. A square lot has an area of 169 sq. rd. How far is it around the lot? How far is it around a lot of four times this area? The second perimeter is how many times the first? Plot each lot to a scale.

33. A man has two adjacent building lots fronting on the street, each lot being square. The area of the two together is 89 sq. rd., that of the larger being 64 sq. rd. What is the frontage of the lots?

34. A square lot has an area of 289 sq. rd. How far is it around the lot? (Try the prime numbers between 10 and 20.) How far is it around a lot of nine times this area? The second perimeter is how many times the first?

565. Cube numbers and cube roots. If a cube has an

edge 3 units, it has a volume 27 cubic units. Therefore 27 is called the cube of 3, and 3 the cube root of 27.

566. Cube roots of volumes. Therefore, considering the abstract num

bers representing the edges and volume,

The edge of a cube is the cube root of its volume.

567. Writing cubes and roots. The cube of 3 is written 38; the cube root of 27 is written

27.

568. Powers. Squares and cubes are called powers. We also have higher powers, like the fourth, fifth, and so on. Raising to powers is sometimes called involution; extracting roots, evolution.

569. Cube roots of perfect cubes. Cube roots of per- 2)216 fect cubes may often be found by factoring.

2)108

For example,

216 =

√2 × 2 × 2 × 3 × 3 × 3

3

2)54

3)27

= √(2 × 3) × (2 × 3) × (2 × 3)

= 6 x 6 x6 = 6.

3)9

3

9. What is the edge of a cube of volume 10,648 cu. in. ?

570. Letters used to represent numbers. If we have two letters, like x and y, the product of their values is indicated by xy. If x = 5 and y = 7, then xy= 5 x 7 = 35, 2xy = 70, x2 = 25, and y2 = 49.

This is all the work with letters necessary for the understanding of square root. If not already known, a few minutes of drill upon similar work will suffice.

280

49

571. Square on the sum of two lines. If we have two lines, f and n, and construct a square on their sum, we see by this figure that there are two squares and two rectangles, f2, n2, and fn, fn. Therefore

The square of the sum of two numbers equals the square of the first, plus twice the

1600

280

product of the first and second, plus the square of the second.

That is,

(f + n)2 = ƒ2 + 2 fn + n2.

572. Illustrative problem. What is the square of 47?

472 = (40+7)2 = 402 + 2 × 40 × 7 + 72