LESSON 184. 1. What is the square root of .0081 ? Ans. .09. Ans. 25.0638, about. 3. What is the square root of 895,372 ? Ans. 946.241, nearly. 4. What is the square root of 14? 5. What is the square root of? 6. What is the square root of 981 ? 7. A square garden contains 6.25 sq. length of one side of it? Ans. 1, or 2. Ans. .433, about. Ans. 9.912, about. rods; what is the Ans. 2.5 rods. 8. A field in the shape of an oblong square, 5 times as long as it is wide, contains 2,672.05 sq. ft.; how many feet wide is it, and what is its length? Ans. width 23.117 ft., about; length 115.585 ft., about. 9. A field containing 61,322 sq. ft. is as wide as it is long; how wide is it, and what is its length? Ans. width 214.456 ft., about; length 285.941 ft., about. Explanation. It is as long as it is wide. 10. What is the square root of .00048 ? Ans. .0219, about. 11. There is a field 20.25 rods square, which I have agreed to exchange for another field equally large, but which shall be three times as long as it is wide; what will be the length and breadth of the field? Ans. 35.073 ft. length, about; 11.691 ft. breadth, about. Figure 21. LESSON 185. If a triangle has a square corner, or right angle, the square of the side opposite the right angle will be equal to the sum of the squares of the two sides adjacent the right angle. For, if the sides adjacent the right angle are equal, as in figure 21, we see that the square of one of them contains two triangles, and is half as large as the square of the side opposite the right angle, which contains four triangles of equal size. If the two sides ad If a triangle has a square corner or right angle, how is the square of the side opposite the right angle to the sum of the squares of the two sides adjacent the right angle? Explain it. jacent the right angle are not equal, it can also be shown that the sum of their squares is equal to the square of the side opposite the right angle. 1. A room is 18 ft. long, and 15 ft. wide; what is the distance between the opposite corners ? Ans. 23.431 ft., nearly. 2. The wall of a fort is 24 ft. high, and there is a ditch beside it 18 ft. wide; how long must a ladder be to reach from the outside of the ditch to the top of the wall? Ans. 30 ft. 3. If the ladder be 20 ft. long, and the wall 16 ft. high, how wide is the ditch? Ans. 12 ft. 4. If the ladder be 25 ft. long, and the ditch 15 ft. wide, how high will the wall be? Ans. 20 ft. 5. A certain field, in the shape of an oblong square, is 40 rods long, and 36 rods wide; what is the distance between the opposite corners ? Ans. 53.814 rods, about. 6. If you make a square with one side 6 ft. long, and the other 8 ft. long, what will be the length of a pole that will measure the distance from the end of one side to the end of the other? Ans. 10 ft Note. Carpenters usually make a square by fastening two pieces of wood together, one 6 ft. long, and the other 8 ft. long, and making the distance between the two ends 10 ft. 7. A carpenter building a house 24 ft. wide, wishes to have the gable end 12 feet high; how long must the rafters be? Ans. 16.97 ft., Explanation. Make a figure of it on your slate. about. 8. What is the distance between the opposite corners of a square field, containing 2 A. 1 qr. 32 sq. rods ? Ans. 28 rods. 9. I wish to hew the largest square stick possible out of a log 16 in. in diameter; what size will the end of the square stick be? Ans. 11.3 in. square, about. Explanation. The diameter of one end of the log is the distance between the opposite corners of the square. 10. If you have one pole 20 ft. long, and another 12 ft., how long must a third be, so that they may form a right angle when put together in the form of a triangle, the 20 ft. pole being opposite the right angle ? Ans. 16 ft. CUBE ROOT. LESSON 186. If we multiply a number and its square together, the product is called the cube of that number; thus, 8 is the cube of 2, 27 is the cube of 3, &c. That number, which, multiplied by its square, will produce a certain other number, is called the cube root of this other number; thus, 2 is the cube root of 8, 3 the cube root of 27, &c. The cube of a number is easily found, being obtained by multiplying the number and its square together, but it is more difficult to get the cube root of a number; however, where the cube root is a whole number, not exceed ing 10, it is readily found by trying a few times. Examples to be performed in the mind. Of 216? Of 512 ? Of What is the cube root of 343? 125? Of 729 ? Of 64? Of 1,000? Of 27? Of 8 ? When the cube root of a number exceeds 10, this manner of obtaining it is tedious; for instance, if the cube root of 2,197, or of 4,096, is required, we are obliged to try many times before we find a number, which, multiplied by its square, will produce either of them. In order to discover a method of getting the cube root with facility when it exceeds 10, we first observe how we find the cube of a number more than 10, say of 27. To make the operation more plain, we multiply the square of 27 in three parts, as we found it in the Square Root, lesson 180, by 27 in two parts, 20 and 7. What is called the cube of a number? What is the cube of 2? Of3? What is called the cube root of a number? What is the cube root of 8? Of 27? What is said of the ease of finding the cube and cube root of a number? What if the cube root of a number exceeds 10? How do we proceed to discover a method of getting the cube root with facility, when it exceeds 10? Therefore, the cube of a number containing two figures, consists of, The cube of the tens, three times the square of the tens multiplied by the units, three times the tens multiplied by the square of the units, and the cube of the units. Now the cube of 10 is 1,000, the cube of 100 is 1,000,000, the cube of 1,000 is 1,000,000,000, &c. It appears, then, that the cube of units is found in the three right hand figures, because the cube of 10, or 1,000, is the smallest number possible consisting of four figures; the cube of tens is found in the three next figures, because the cube of 100, or 1,000,000, is the smallest number possible consisting of seven figures. It can be shown in the same way, that the cube of hundreds is found in the three figures at the left of the cube of tens; that the cube of thousands is found in the three figures at the left of the cube of hundreds, &c. Let us find the cube root of 19,683. Explain what the cube of 27 consists of? What then does the cube of a number, containing two figures, consist of? Where is the cube of units found? Why? Where is the cube of tens found? Why? What else can be shown? Explanation. The cube of the units is in 683, and the cube of the tens in 19. The greatest cube in 19 is 8, the cube root of which is 2; this root we place at the right, like a quotient; subtract 8, the cube of 2, from 19, and bring down the 683. The remaining number, 11,683, contains 3 times the square ofthe tens multiplied by the units, 3 times the tens multiplied by the square of the units, and the cube of the units; now 3 times the square of any number of tens multiplied by units, gives nothing less than hundreds; so if we divide the 116 hundreds in 11,683 by 3 times the square of the 2 tens, or 1,200, we shall get the number of units, or too large a number, since 3 times the tens multiplied by the square of the units, and the cube of the units, usually increase the hundreds considerably. In fact, we get 9, which, on the proper trial, is found too large, as well as 8. Let us try 7; placing it at the right of 2, to find whether it is right or not, we add together 3 times the square of the tens multiplied by 7, 3 times the tens multiplied by the square of 7, or 49, and the cube of 7, or 343; the sum making 11,683, the same as the remaining number, we conclude that 7 is right, and that 27 is the cube root of 19,683. Explain how you find the cube root of 19,683. |