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; then +16=the length; and x16x the number of square yards. Hence + 16 r = 960. 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 x + a, is x2, +2 a x + a2, which consists of three terms. Now if we compare this with the first member x2 + 16 x, we find (x+8) (x+8) = x2 + 16 x + 64. Hence, if to x2 + 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 x2+16x+64 = 960 + 64 = 1024. Taking the root of both members x+8=± (1024)* = 32. It has been already remarked that the 2d root of every positive quantity, may be either positive or negative, because a x aa as well as a x + a + a2. double sign is read plus or minus. The 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. x+8=±32. This gives the length 40 yards and the breadth 24. These numbers answer the conditions of the question. 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 × — 40 + 16 × — 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 1 shilling apiece more than they would have done, if all had paid. What was the whole number of persons? 143 the number after 4 have departed; = the number of shillings each should have paid; Clearing of fractions 143x+x2 572-4x=143 x By transposition x2-4x572. This equation is similar to the last, except in this, the second term of the first member has the sign Here we must observe that the second power of the binomial a, is x2 2 a x + a2, the same as that of x+a with the exception of the sign of the second term. In this equation, as before, we find two terms of the second power of a binomial ; if we can find the other term we can easily solve the question. Adding 4 to both members of the equation it becomes · 4 x + 4 = 572 + 4 = 576. Since -2 in this corresponds to a, the root of the first mem ber is x-2. In fact, (x-2) = x2-4x+4. The root of 576 is 24. Hence x-2= 24 x=224. The two values of x are 26 and -22. The former only an swers the conditions of the question. Proof. If the whole number, 26, had paid their shares, each would have paid 45 shillings. But 22 only paid, consequently each paid 36 shillings. 26 143 22 3. There are two numbers, whose difference is 9, and whose sum multiplied by the greater produces 266. What are those numbers? |