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8. What is the third root of 17 ?
The fractional part of this number must first be changed to a decimal. 171 = 17.75 = 477570
67° = 17.750. Hence it appears, that to prepare a number containing decimals, it is necessary that for every decimal place in the root, there should be three decimal places in the power. Therefore we must begin at the place of units, and separate the number both to the right and left into periods of three figures each. If these do not come out even in the decimals, they must be supplied by annexing zeros to the right.
9. What is the approximate third root of 25732.75? 10. What is the approximate third root of 23.1762 ? 11. What is the approximate third root of 12;? 12. What is the approximate third root of 111? 13. What is the approximate third root of ij? 14. What is the approximate third root of ??
XXXIII. Questions producing Pure Equations of the Third
1. A man wishes to make a cellar, that shall contain 31104 cubic feet; and in such a form, that the breadth shall be twice the depth, and the length 13 the breadth. What must be the length, breadth, and depth ?
Let the depth =X, the breadth = 2 x,
and the length =$1
2. There are two men whose ages are to each other as 5 to 4, and the sum of the third powers of their ages is 137781. What are their ages?
Ans. Elder 45 years, and younger 36. 3. A man wishes to make a cubical cistern that shall contain 100 gallons. What must be the length of one of its sides?
4. A bushel is 2150 cubic inches. What must be the size of a cubical box to hold i bushel ?
5. What must be the size of a cubical box to hold 2 bushels ?
6. What must be the size of a cubical box to hold 8 bushels ?
7. Find two numbers, such that the second power of the greater multiplied by the less may be equal to 448; and the second power of the less multiplied by the greater, may be 392 ?
8. A man wishes to make a cistern which shall hold 500 gallons, in such a form that the length shall be to the breadth as 5 to 4, and the depth to the length as 2 to 5. Required the length, breadth, and depth.
Note. The wine gallon is 231 cubic inches.
9. A man wishes to make a box which shall hold 40 bushels, in such form that the length shall be to the breadth as 4 to 3, and the depth to the breadth as 2 to 3. Required the length, breadth, and depth ?
10. A man bought a piece of land for house lots, the breadth of which was to its length as 3 to 28; and he gave as many dollars per square rod, as there were rods in the length of the piece. The whole price was $63,504. Required the length and breadth.
11. A man agreed to sell a stack of hay for 10 times as many dollars as there were feet in the length of one of the longer sides. On measuring it, the length was to the breadth as 6 to 5, and the breadth and height were equal. Moreover it was found that it came to as many cents per cubic foot as there were feet in the breadth. Required the dimensions of the stack.
XXXIV. Affected Equations of the Second Degree. When an equation of the second degree consists only of terms which contain the second power of the unknown quantity, and of terms entirely known, they may be solved as above. But an equation of the second power, in order to be complete, must contain both the first and second powers of the unknown quantity, and also one term consisting entirely of known quantities. These are sometimes called affected equations.
1. There is a field in the form of a rectangular parallelogram, whose length exceeds its breadth by 16 yards, and it contains 960 square yards. Required the length and breadth.
Let x = the breadth ;
In order to solve this equation, it is necessary to make the first member a perfect second power.
Observe that the second power of the binomial é + a, is z", + 2 a x ta', which consists of three terms.
Now if we compare this with the first member ** + 16 x, we find
2 a x = 16 x which gives
2a = 16 and
a = 8
a = 64
(x-+ 8) (+ 3) = x + 16 x + 64. Hence, if to r* + 16 x we add 64, which is the second power of one half of 16, the first member will be a perfect second power, but it will be necessary to add the same quantity to the second member, in order to preserve the equality. The equation then becomes
xos + 16 x + 64 = 960 + 64 = 1024. Taking the root of both members
*+8= + (1024))= 32. By transposition X=-832.
It has been already remarked that the 2nd root of every positive quantity, may be either positive or negative, because a X - a= ta' as well as taxta= ta. The double sign is read plus or minus.
In the preceding examples, the conditions of the question have always determined which was to be used. But, in the present instance, the work not being completed when the root is taken, we must give it both signs, and when the values of x are found for both signs, the conditions will finally show which is to be used.
+8 = 32.
If we use the sign +, we have
X = 24
X + 16 = 40. This gives the length 40 yards and the breadth 24. These numbers answer the conditions of the question. If we use the sign
-40 i + 16
- 24. These numbers will not satisfy the conditions of the question, but they will answer the conditions of the equation, as will be seen by putting them into the first equation.
- 40 + 16 x .40 = 960. 2. A certain company at a tavern had a reckoning of 143 shillings to pay ; but 4 of the company being so ungenerous as to slip
away without paying, the rest were obliged to pay. I shilling apiece more than they would have done, if all had paid. What was the whole number of persons ? Let x = the number of persons at first ;
4= the number after 4 have departed ; 143
= the number of shillings each should have paid ;
Clearing of fractions
143 x + x— 572 - 4x = 143 x By transposition
2 - 4x = 572.