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87. Roots of Numbers Greater than 1000.-For getting the cube roots of numbers greater than 1000, the easiest and most accurate way is to look in the third column headed "cubes" for a number as near as possible to our given number. Now, we know that the numbers in the first column are the cube roots of these numbers in the third column. If we can find our number in the third column, there is nothing further to do because its cube root will be directly opposite it in the first column. Examples:

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Likewise, the numbers in the first column are the square roots of the numbers in the second column. But suppose we want the cube root of a number which is not found in the third column, but lies somewhere between two consecutive numbers in that column. In this case we pursue the method shown in the following example:

Example:

Find /621723

In the column headed "Cube" find two consecutive numbers, one larger and one smaller than 621723. These numbers are

636056 whose cube root is 86

and 614125 whose cube root is 85

Hence, the cube root of 621723 is more than 85 and less than 86; that is, it is 85 and a decimal, or 85+.

The decimal part is found as follows: Subtract the lesser of the two numbers found in the table from the greater and call the result the First Difference.

636056-614125=21931, First difference.

Then subtract the smaller of the two numbers in the table from the given number and call the result the Second Difference.

621723-614125=7598, Second Difference

21931

Now the first difference, 21931, is the amount that the number increases when its cube root changes from 85 to 86. Our given number is only 7598 more than the cube of 85, so its cube root will be approximately 85& We do not want a fraction like this, so we reduce it to a decimal as follows: Divide the second difference by the first difference and annex the quotient to 85. This will give us the cube root of our number, approximately.

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This is the decimal part of the root sought and the whole root is 85.346+. Hence /621723=85.346+.

This method is not exact and the third decimal place will usually be slightly off, so it is best to drop the third decimal if less than 5, or raise it to 10, if more than 5. In this case we will call the root 85.35.

88. Cube Roots of Decimals.-In getting the cube root of either a number entirely decimal, or a mixed decimal number, it is best to move the decimal point a number of periods, that is, 3, 6, 9, or 12 decimal places, sufficient to make a whole number out of the decimal. After finding the cube root, shift the decimal point in the root back to the left as many places as the number of periods that we moved the decimal point in our original number. For example, suppose that we had .621723 of which to find the cube root. Moving the decimal point two periods (of three places each) to the right gives us 621723, of which we just found the cube root to be 85.35. We moved the decimal point of our original number two periods to the right, so we must move the decimal point back two places to the left in the root; we then have .621723.8535. The following illustrations will show the principle:

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Notice that there is no such similarity between the cube roots of numbers if we move the decimal point any other number of places than a multiple of three.

But

=

=

V/6 1.817+ /60 3.915 6000=18.17 /60,000=39.15

/600 8.434 600,000=84.34

Care should be taken, therefore, that, if necessary to move the decimal point in finding a cube root, it should be moved an exact multiple of 3 places. If we have a decimal such as .07462, it is necessary to attach a cipher at the right, making the decimal .074620, so we can shift the decimal point 2 periods or six places. We can now find 74620 as follows:

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89. Square Root by the Table.-The same methods as have been explained for cube root can be applied to finding square roots by the use of the table. The only difference is in the case of decimals, in which case the decimal point is shifted by multiples of two places in the number; then, after we have the root; we shift the decimal point back one place for each period of two places that we moved the decimal point in our original number.

PROBLEMS

The tables are to be used wherever possible in working these problems. 151. What would be the length of a steel sheet from which to make a 28 in. circular drum, allowing 1 in. extra for lapping and riveting the ends? The circumference of the drum is measured in the direction of the length of the sheet.

152. The small sprocket of a bicycle contains 8 teeth, the large sprocket 24 teeth. The rear wheel is 30 in. in diameter. Find the distance travelled over by the bicycle for one revolution of the pedals.

153. The diameter of a 13 in. bolt at the bottom of the threads is 1.16 in. What is the sectional area at the bottom of the threads?

154. A No. 000 copper trolley wire has a diameter of 0.425 in. What would be the cost of 1 mile of the wire at 33 cents a pound?

09

∙14

FIG. 35.

155. A circular piece of boiler plate (Fig. 35) } in. thick and 60 in. diameter has an elliptical man hole in it 14 in. by 10 in. Find weight of plate. Note.-Area of an ellipse =.7854 Xa Xb, where a and b are the long and short diameters of the ellipse.

156. The paint shop wants a cubical dip tank built to hold, when full, 400 gallons of varnish. What will be the dimensions of the tank? (There are 231 cu. in. in a gallon.)

157. How many square feet of galvanized iron will be needed to line the tank of problem 156 on four sides and the bottom, allowing 10% extra for the joints?

158. A high carbon steel contains the following items in the percentages given: Carbon, .60%; silicon, .10%; manganese, .40%; phosphorus, .035%; sulphur, .025%. The rest is pure iron. Calculate the weights of carbon, silicon, manganese, phosphorus, sulphur, and iron in one ton (2000 lb.) of the steel.

159. I want to get a cast iron block 18 in. long and of square cross-section so that it will weigh 200 lb. How many cubic inches of metal must be in the block and what will be its dimensions?

If we

160. Fig. 36 shows a steam hammer having an 8000 lb. ram. assume that this ram is a rectangular block of steel, four times as high and twice as wide as it is thick, what will be the dimensions of the ram?

FIG. 36.

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