41. INDIRECT PROBLEMS IN MEASUREMENT

1. What is the side of a square whose area is 81 sq. in.?

SOLUTION: Since 81, the number of square inches, is the product of two equal numbers, each of which is the number of inches in the length of the square, we know that the answer is 9 in., for 9 x 9 = 81.

2. What is the side of a square whose area is 1225 sq. in.?

3. What is the side of a square having the same area as a rectangle 16 ft. wide and 36 ft. long?

SOLUTION: 16 x 36 = 4 × 4 × 6 × 6. Hence the square is 24 in. long. 4. A lot 75 ft. wide and 147 ft. long has the same area as a square lot of what dimension ?

5. What is the radius of a circle whose area is 314.16 sq. ft.?

SOLUTION: The area is the product of two factors. One is R2 and the other is 3.1416. Dividing by 3.1416, the known factor, the other factor is found to be 100. Hence R2 = 100. Then R = 10.

6. What is the radius of a circle whose area is 154 sq. in.? SUGGESTION. Use T = 22. First divide 154 by 27.

7. What must be the radius of a circular bottom of a silo if its area is to be 616 sq. ft.? (Use T = 22.) The diameter?

Find the two equal factors that make the following:

42. THE NEED OF A NEW PROCESS

In the problems of the preceding page it was found that it sometimes is necessary to find one of two equal factors that make a number.

When a number is the product of two or more equal factors, it is called a power of one of the factors. When it is the product of two equal factors, it is called the second power or square of one of them, and when the product of three equal factors, it is called the third power or cube of one of them. The power of a number is indicated by an exponent. Thus, 2 × 2 × 2 is written 23, where 3 is the exponent.

When a number is the product of two or more equal factors, one of the factors is called a root of the number. If there are two equal factors, one is called the square root, and if there are three equal factors, one is called the cube root of the number.

The square root of a number is indicated by the symbol √, called a radical sign. Thus, the square root of 25 is written √25.

1. Find the radius of a circle whose area is 300 sq. ft.

=

SOLUTION: 300 ÷ 3.1416 95.5, nearly. Now 95.5 is less than 10 x 10 and greater than 9 x 9. Also, since 9.5 × 9.5 = 90.25, 95.5 is also greater than 9.5 x 9.5. Then the radius is greater than 9.5 ft. but less than 10 ft. 2. Show that the square root of 240 is greater than 15.4 and less than 15.5.

While the approximate results found above might be sufficient for many practical problems, we need a new process by which more accurate results may be found when wanted. This new process is called Square Root. While the exact square root of a number cannot be found unless the number is made up of two equal factors, the process gives the root to any desired degree of accuracy.

43. THE PROCESS OF EXTRACTING THE SQUARE ROOT OF ANY NUMBER

Separating a number into two equal factors is the reverse of squaring one of these equal factors. A careful analysis of the process of squaring will enable us to reverse the process and find the square root of a number when it cannot be readily found by factoring.

= 2 × 3 × 80. Hence, 832 = 32 + 2 × 3 × 80 + 802. 4. Square 64 by this method.

WORK

64

64

16= 42

5. Which of the partial products is the largest? From which digit was it obtained?

6. Which is the smallest of the par

480 = 2 × 4 x 60 tial products? From which digit was 3600 = 602

4096

642

it obtained?

7. If 3600 were taken from the prod

uct, most of what remains is made from what factors? 8. State the rule for squaring numbers in this way. Find by this method the squares of the following:

9. 35.

10. 62.

11. 84.

12. 58.

Comparing the Number of Figures in Roots and Powers 1. Give the squares of all the numbers from 1 to 9 inclusive.

2. How many figures in each of these squares?

3. Square the numbers 10, 20, 30, and so on to 100. 4. How many figures in the squares of numbers from 10 to 99 inclusive?

5. What is the square of 100? Of 200? Of 999?

6. How many figures in the squares of numbers from 100 to 999 inclusive?

7. If there are four figures in the square, how many in the root? How many in the root when five figures are in the square?

From Exercises 1-6 we see that the squares of the smallest and the largest integers composed of one, two, or three figures are as follows:

12=1 9281

102=100
992 =
= 9801

1002 = 10,000
9992998,001

8. Separate each of the squares shown above into periods of two figures each, beginning at the right. Thus, 98' 01'; 1' 00' 00'; etc.

9. Compare the number of periods in each square with the number of figures in the corresponding root.

The number of periods of two figures each, beginning at ones, into which a whole number can be divided equals the number of figures in the square root.

10. Give the number of figures in the square roots of: 9409 381 27,225 182,329 49,434,961. 11. Square 0.2; 0.02; 0.4; 0.12; 0.25; 0.03; 0.005.

12. Compare the number of decimal places in the square with the number in the root. Why can the square of a decimal never contain an odd number of decimal places?

13. State a principle for the number of periods in the decimal part corresponding to the principle given for integers.

Extracting the Square Root

Find the square root of 2809.

300 2 x 50 x 3

=

9 = 32

3. What is the square of 50? Of 60?

4. Between what two squares does 2809 come?

5. Then its root lies between what two numbers?

6. If the root lies between 50 and 60, the largest of the three partial products that make the square is what?

7. When 2500 is taken from 2809, what two partial products are contained in the 309 remaining? Most of the 309 is made from which of the partial products?

8. Then since 309 is more than 2 x 50 times the number that we are yet to find, about what must the number be?