57. Show that the area of the picture frame represented in Figure 9 is A-ld+wd-4d2d(l+w-4d). Find A

FIG. 9

d

d

Find r

59. Solve c=2πr for r. correct to .01, if c=120 ft. Use T=34.

60. Make a formula which gives the perimeter, p, of an isosceles

triangle one of whose equal sides is 7 and whose base is b. Solve this formula for l. Find l if p=90 ft. and b=18 ft.

61. Solve for c, c+.25c-875.

62. Solve for c, c-15% of c=3.91.

63. Solve for s, s-1.2% of s=829.92.

64. By the breaking strength of a rope is meant the least pull that is likely to break it. The breaking strength of manila rope is 3000 lb. per square inch of cross section area. Make a formula stating this fact, where I is the breaking strength and d the diameter of the rope.

65. What weight would be likely to break a manila rope 1 in. in diameter? One 1 in. in diameter?

CHAPTER III

SQUARE ROOT AND ITS APPLICATIONS

25. To find the length of one side of a square whose area is known. The area of a square is 9 sq. in. How are these 9 square inches arranged?

Since the number in a row is the same as the number of rows, we must find the number whose square is 9. This is 3. Therefore there are 3 rows of square inches, each row containing 3 square inches. Therefore each side of the square is 3 inches long.

The number whose square is 9 is called the square root of 9. It is expressed thus: √9. The symbol is called a radical sign.

The square root of a number is one of the two equal factors of the number.

Thus, 5 is the square root of 25; 8 is the square root of 64; and 10 is the square root of 100.

EXAMPLE. Find the length of one side of a square whose area is 49 square inches.

SOLUTION. The area is equal to 49 sq. in.

Therefore the length of one side = √49 inches. √49=7.

Therefore the length of one side =7 inches.

In finding the square roots of numbers it is useful to know the squares of the integers from 1 to 25 inclusive. The pupils should commit them to memory and be able to repeat them in 1 minute.

25. 81 sq. rd.

Find the length of the side of the square whose area is :

26. To find the square root of larger numbers by factoring. If the square root of 36 were forgotten it could be found thus: 36 2.18 2.3.6 2.3.3.2= (2.3) (2·3)=62.

Therefore 36=6.

=

Rule. Break the number up into factors, then arrange these factors, if possible, into two groups having the same factors. The product of the factors of one of these groups is the square root of the number.

The solution may be shortened if the pupil sees that 441 equals 21.21. He may then write

1764 2.2.21·21 = (2·21) (2·21) =422.

Exercise 28

Find by factoring the square roots of the following:

27. The square of the sum of two numbers.

1. Write the sum of a and b.

2. Indicate the square of the sum of a and b. 3. By multiplying find the square of a+b.

EXAMPLE 1. (x+y)2=?

(x+y)2=(x+y)(x+y)

= x2+2xy+y2

4. (m+n)2=? Find by multiplication.

x + y
x + y

x2+xy

xy+y2

x2+2xy+y2

5. (s+t)? Write the result without multiplication. These illustrate the principle: The square of the sum of two numbers equals the square of the first number plus two times the product of the two numbers plus the square of the second number.

Stating this principle as a formula,

(a+b)2=a2+2ab+b2.

EXAMPLE 2. Find the square of the sum of 9 and 6.

Exercise 29

1. Indicate the sum of n and s; the square of n and s.

of the sum

2. (a+2)2 indicates the square of the sum of what two numbers?

3. (4) indicates the square of the sum of what two numbers?

Using the principle above find the squares of the following

7. (5+7)2. 11. 452=(40+5)2= ? 15. (61)2. 16. (3)2. 17. The expression a2+2ab+b2 is the square of the sum of what two numbers? How is the first number found? How can the second number be found? Answer the same questions for x2+2xy+y2.

18. m2+2mn+n2=(?+?)2. Check by squaring the result. 19. 42+2.5.4+52=(?+?)2.

20. 4x2+20xy+25y2=(?+?)2.

21. In the expression a2+2ab+b2, 2ab is twice the product of the two numbers whose sum was squared. What can be done to 2ab to find the product of the two numbers? If you knew one of the numbers how could you then find the other?

22. In the expression 32+30+(?)2, 30 is twice the product of the two numbers whose sum was squared. What is their product? What is the first number? How can you find the second? Supply the missing third term of the expression. Supply the missing numbers in these equations:

23. 32+2.3.4+(?)2=(?+?)2.

24. 25+10.3+(?)2=(?+?)2. 25. 36+60+(?)2 = (?+?)2.