29. Find the areas of fields the dimensions of which are given below in links. 3rd Left Right 246 125 162 372346 223 318 Base line A 266 345 465 560 718 790 987 1015 1325 B 134 58 136 Ans. 5 acres 36 perches. O O 1253 B B 200 400 600 846 C O 50 95 64 Ans. 2 acres 3 roods 20 perches. 246 O 30. ABC be a right-angled triangle, in which B is the right angle, Ans. 55. Ist. Find A C when A B=33 ft., BC=44 ft. Ans. 21 73. 4th. FindA C when A B=4.5 yds., BC=9yds. Ans. 10·062. 31. A ladder 46 ft. in length, being placed in a street, reached a window 26 ft. from the ground on one side; and by turning it over, without removing the bottom, it reached another window 35 feet high, on the other side; what is the breadth of the street? Ans. 67.79695 ft. 32. Two travellers, A and B, departed from an inn, at the hour of eight in the morning; A proceeded north-west, at the rate of 6 miles an hour, and B north-east, at the rate of 8 miles an hour; how far were they distant from each other, at twelve o'clock of the same day? Ans. 40 miles. 33. A ladder and a wall are each 78 ft. high. If the ladder be placed against the wall, and then its foot be drawn out 15 ft., how far will the top descend? 34. The diagonal of a square is 6 ft.; find its area. Ans. 18 square feet. 35. A field in the form of a right-angled triangle is to be divided between two persons, by a fence made from the right-angle meeting the hypothenuse perpendicularly, at the distance of 880 links from one end; required the area of each person's share, the length of the division-fence being 660 links. Ans. 2ac. 3roods 24 per. and I ac. 2 roods 211 per. 36. There are two columns in the ruins of Persepolis left standing upright, one of which is 64 feet above the horizontal plane, and the other 50. Between these, in a right line, stands a small statue, the head of which is 97 ft. from the summit of the higher, and 86 feet from the top of the lower column, whose base is just 76 ft. from the centre of the figure's base; required hence the distance between the tops of the two columns. Ans. 157.03687 ft. 37. A gentleman has a garden 100 ft. long and 80 ft. broad, and a gravel walk is to be made of an equal width half round it ; what must the breadth of the walk be, to take up just half the ground? Ans. 25.968 ft. 38. The top of a maypole being broken off by a blast of wind, struck the ground at 10 ft. distance from the foot of the pole ; what was the height of the whole maypole, supposing the length of the broken piece to be 26 ft.? Ans. 50 feet. CHAPTER VII. Proportion of Numbers-Principles of Proportion. 166. Suppose you buy 4 yds. of cloth for 12s., 8 yds. of the same will evidently cost 245., 2 yds. would cost 6s., and so on. By whatever number we multiply the 4 yds., by the same number must we multiply the 12s. to obtain the price. This is expressed by saying that the price of the cloth is proportional to its length, or, that the numbers 4, 8, 12, and 24 form a proportion. The first number of yards, 4, is to the second, 8, as the first cost price, 12s., is to the second, 24s.; or again, the ratio of the two numbers of yards is equal to the ratio of the two corresponding numbers of shillings. Whatever may be the nature of the magnitudes represented by four numbers, if the ratio. of the first pair is the same as the ratio of the second, it is said that the four numbers are in proportion, or that they form a proportion. We have already said that the ratio of two numbers is the quotient of the first divided by the second. Thus the ratio of two numbers, 56 and 7, being equal to the ratio of 48 and 6, since the quotient obtained by dividing 56 by 7 is 8, and that obtained by dividing 48 by 6 is also 8, the numbers 56, 7, 48, 6, form a proportion. To say, then, that the ratio of two numbers is equal to the ratio of two other numbers, or that the four numbers form a proportion, is to say the same thing in different. words. The ratio of two numbers is represented by putting the first above the second, and separating them by a horizontal line; the ratio of 56 to 7 is written and 7 are called the terms of the ratio; numerator and 7 is the denominator. 56 7 ; 56 56 is the We have then, the equation 56 48 which signifies that 56 divided by 7 gives the same quotient as 48 divided by 6. We may make the same statement thus Or again 56 is to 7 as 48 is to 6 56 7 48: 6 The four numbers are called the terms of the proportion; 56 and 6 are called the extremes, 7 and 48 the means. 167. If we multiply 56 by 6 we obtain 336; if we multiply 7 by 48 we also obtain 336; and the equality of the products in this particular example is enunciated as a general principle when we say In every proportion the product of the extremes is equal to the product of the means. H In order to see that this property does not belong only to particular numbers, write the proportion under 56___ 48 its first form = 7 6 Since 56 contains as many times 7 as 6 times 56 56×6 7 X6 contains 6 times 7, the first ratio is equal to Again; 48 contains as many times 6 as 7 times 48 contains 7 times 6; the second ratio may therefore be re 48 × 7 placed by and we may write the equation 56 × 6 thus 48 X 7 = 7×6 6×7 The divisors in these two equations are equal, so, therefore, are the dividends; thus, 56 X 6 48 x 7. X = 168. Conversely; If four numbers are such that the product of two of them is equal to the product of the remaining two, these four numbers form a proportion; for example, the four numbers 5, 54, 6, 45 are such numbers since 5 X 54 6 × 45; therefore they form a proportion, for dividing these two equal numbers by the product 5 × 6, the quotients will be equal. We 5 × 54 6 × 45 5 × 6 5×6 shall have then = ; but a process of reasoning similar to that already given, will show that we may divide the two terms of the first ratio by 5, and the two terms of the second by 6, without altering their value; we have then— = 54_45 ', or 54: 6:: 45: 5. 169. The fourth term of a proportion is equal to the product of the means divided by the first term. = = For since 54 X 5 6 x 45, or 54 times 5 6 × 45, once 5 is found by dividing this product by 54, or 6 × 45 54 170. Three terms of a proportion being given, to find the fourth. Let the three given terms be 12, 108, and 11. Let us mark the fourth term by x— 12: 108 :: II : x And from what has already been said— Again, take the three numbers 4, 6, and 6, we have 6×6 4: 6 :: 6 : x; whence x = = 9 4 the proportion is then 4 6:: 6 : 9. When the two means are equal, the number x is called a THIRD PROPORTIONAL, and the mean is termed the geometrical mean of the two extremes; thus 6 is called the geometrical mean between 4 and 9. We have said that from the equation we can conclude that the four numbers 5, 54, 6, 45 form a proportion (§ 168). Now this proportion may be written in the four following ways: 54 45 45 56/56 = 45 54 For the product of the two extremes remains always equal to the product of the means. in every proportion we may invert Ist. The two means only. 2nd. The two extremes only. 3rd. The two means as well as the two extremes. |