3d. Suppose a = b. to In this case, the first value of x is positive, and equal to Hence the first point of equal illumination is midway between A there is no other point of equal illumination in the line AB, or in AB produced, at a finite distance from A. These conclusions are obviously correct. For, under the present supposition, the two lights are equally intense. Hence any point, to be equally illuminated by them, must be equally distant from them; and the only point which fulfills this condition is the point midway between them. If, however, we consider a and b as two varying quantities, at first unequal, but continually approaching equality, then the second value of x will become greater and greater by degrees, until it reaches infinity. Under these conditions, the second point of equal illumination will continually recede from A, moving toward the right or toward the left, according as a is greater, or less than 6, until it is finally removed to an infinite distance. In this view of the case, it is sometimes said that there are two points of equal illumination, under the hypothesis, a = b; one point being at an infinite distance from A. (188, 4). This result shows that there are an infinite number of other points equally illuminated by the two lights. These interpretations are evidently correct. For, as the lights, under the present hypothesis, are equally intense, and both situated at A, every point in space must be equally illuminated by them 5. Suppose c = 0, and a> b or ab. Both values of x now reduce to 0; and the common rule for interpreting zero might lead us to suppose that the two points of equal illumination coincide with the point A. But this conclusion is not strictly correct; for it is obvious that when two lights, of unequal intensities, occupy the same place, there is no point in space equally illuminated by them; not even the point in which they are both situated. Let us return to the original equation (m), which truly represents the conditions of the problem. If we put c= = 0, the result is an equation which can not be satisfied by any value of x whatever, while ab or a b. For by substituting any value for x we shall always obtain two unequal fractions. If x = 0, the two members are two unequal infinities. We conclude, therefore, that under the supposition, c = 0, while a and b are unequal, the problem fails altogether, and is impossible. Thus we learn that zero may be the answer to a possible, or an impossible problem. And whenever we obtain this symbol as the result of a solution, we must not interpret it on the assumption that the thing required in the problem is possible; but we must first determine whether the conditions are rational or absurd, by considcring the nature of the problem, or by substituting zero in the original equation. PROBLEMS PRODUCING QUADRATIC EQUATIONS. 314. It will be found that some of the following problems may be solved by a single unknown quantity, while others require two. Still others may be conveniently solved by means of either one or two letters. It is left to the judgment and skill of the learner to discover the mode of solution, in each example, which is most simple. be 1. It is required to divide the number 14 into two such parts, that 9 times the quotient of the greater divided by the less, may equal to 16 times the quotient of the less divided by the greater. Ans. 8 and 6. °+20= 350 2. A company, dining at an inn, agreed to pay $3.50 for the entertainment; but before the bill was presented, two of the party 23- left, in consequence of which each of the others had to pay 20 cents more than if all had been present. How many persons dined? Ans. 7. 3. There is a certain number, which being subtracted from 22, 2-x)x=117 and the remainder multiplied by the number, the product will be 117. What is the number? Ans. 13 or 9. 4. It is required to divide the number 18 into two such parts, that the squares of these parts may be to each other as 25 to 16. Ans. 10 and 8. 4118 5. The difference of two numbers is 4, and their by the difference of their second powers, is 1600. - = 44 = 96 numbers? (24-48 (x-2)=1600 7x+34 3x2=504 10 4 sum multiplied x + 2 What are the Ans. 12 and 8. 6. What two numbers are those whose difference is to the less as 4 to 3, and whose product multiplied by the less is equal to 504? Ans. 14 and 6. 3 7. A man purchased a field, whose length was to its breadth as 8 to 5. The number of dollars paid per acre was equal to the number -ac of rods in the length of the field; and the number of dollars given for the whole was equal to 13 times the number of rods round the field. Required the length and breadth of the field. Ans. Length, 104 rods; breadth, 65 rods. 4,44, cents per 0x at that rate 224 times as many cents as there are square feet on the XK4=224 bottom. Required the dimensions of the stack. x2=16 Ans. Length, 20 feet; breadth, 16 feet; height, 14 feet. X,100-x 18 NOTE.-Represent the number by -7. 10. A and B together carried 100 eggs to market, and each received the same sum. If A had carried as many as B, he would 100 have received 18 pence for them; and if B had taken as many as 11. The sum of two numbers is 6, and the sum of their cubes is 12. A man traveled 36 miles in a certain number of hours; if he had traveled one mile more per hour, he would have required 3 hours less to perform his journey. How many miles did he travel per hour? 36 36 +3 Y+1 x2+4=12 Ans. 3 miles. 13. The sum of two numbers is 100, the difference of their square roots is two; what are the numbers? Ans. 36 and 64. 14. A gentleman bought a number of pieces of cloth for 675 dollars, which he sold again at 48 dollars a piece, and gained by the bargain as much as one piece cost him. What was the number of pieces? 4=6754 46 Ans. 15. 39 Ans. $30. 16. A merchant sent for a piece of goods and paid a certain sum for it, besides 4 per cent. for carriage; he sold it for $390, and and thus gained as much per cent. on the cost and carriage as the 12th part of the purchase money amounted to. For how much did he buy it? 37 2 264 Ans. $300. 17. From two towns, 396 miles apart, two persons, A and B, set out at the same time, and traveled toward each other; after as many days as are equal to the difference of miles they traveled per day, they met, when it appeared that A had traveled 216 miles. How many miles did each travel per day? 185 221 Ans. A, 36; B, 30. 18. Divide the number 60 into two such parts that their product shall be 704. =704 Ans. 44 and 16. 19. A vintner sells 7 dozen of sherry and 12 dozen of claret for £50, and finds that he has sold 3 dozen more of sherry for £10 than he has of claret for £6. Required the price of each. R น 1 643 Ans. Sherry, £2 per dozen; claret, £3. 20. A set out from C towards D, and traveled 7 miles a day. 184 After he had gone 32 miles, B set out from D towards C, and went +7x+ 3 2 every day's of the whole journey; and after he had traveled as many days as he went miles in a day, he met A. Required the distance from C to D. Ans. 76 or 152 miles. 24 x+16 +18 $ +16 21. A farmer received $24 for a certain quantity of wheat, and an equal sum at a price 25 cents less per bushel for a quantity of 4 barley, which exceeded the quantity of wheat by 16 bushels. How many bushels were there of cach? 4(+-18) 63 4 +18 13 28 1(x+18) Ans. 32 bushels of wheat and 48 of barley. 22. Two travelers, A and B, set out to meet each other, A leaving Cat the same time that B left D. They traveled the direct road, A and met 18 miles from the half-way point between C and D; and it appeared that A could have traveled B's distance in 153 days, and B could have traveled A's distance in 28 days. Required the distance between C and D. Ans. 252 miles. x+23 Find two numbers, whose difference, multiplied by the difference of their squares, is 32, and whose sum, multiplied by the sum of their squares, is 272. 7 Ans. 5 and 3. = x = rate X 2 32 24. A and B hired a pasture at a certain rate per week, agreeing = 13's horsehat each should pay according to the number of animals he should have in the pasture. At first A put in 4 horses, and B as many as cost him 18 shillings a week; afterward B put in 2 additional horses, and found that he must pay 20 shillings a week. At what rate was the pasture hired? Ans. 30 shillings per week. -18:18 11 4 -20:2014. 25. If a certain number be divided by the product of its two 10x+7= digits, the quotieat will be 2; and if 27 be added to the number, x + y +27=/ the digits will be inverted. What is the number? Ans. 36. 26. It is required to find three numbers, such that the difference of the first and second shall exceed the difference of the second and third by 6, the sum of the numbers shall be 33, and the sum of the squares 441. Ans. 4, 13, and 16. 27. What two numbers are those whose product is 24, and whose sum added to the sum of their squares is 62? Ans. 4 and 6. さ |