Hence the area of the base is 45 square feet. EXAMPLES XXXV. c. 2, find 1. If x oc y, = 18, find x when y=21. 2. If x oc y, and y=3 when c= y when x=18. 3. varies jointly as B and C; and A=6 when B=3, C=2: find A when B=5, C=7. 4. A varies jointly as B and C; and A=9 when B=5, C=7: find B when A=54, C=10. 1 5. If xoc and y=4 when x=15, find y when x=6. 7. A varies as B directly, and as C inversely; and A=10 when B=15, C=6; find A when B=8, C=2. 8. If x varies as y directly, and as z inversely, and x=14 when y=10, z=14; find z when x=49, y=45. 1 9. If.coC and z , prove that z of x. 10. If a cc b, prove that an cc bn, 11. If xoc and y oc 2, prove that x2 - y2 o 22. 12. If 3a+70 oc 3a +136, and when a=5,b=3, find the equation between a and b, 13. If 5x – y oc 10x – 1ly, and when x=7, y=5, find the equation between x and y. 14. If the cube of x varies as the square of y, and if x= 3 when y=5, find the equation between x and y. 15. If the square root of a varies as the cube root of b, and if a=4 when b=8, find the equation between a and b. 16. If y varies inversely as the square of x, and if y=8 when x=3, find x when y=2. 17. If xocy+a, where a is constant, and x=15 when y=1, and x=35 when y=5; find x when y=2. 18. If a +boca-b, prove that a? + b2 oc ab; and if a oc b, prove that a2- 62 oc ab. 19. If y be the sum of three quantities which vary as x, x2, 28 respectively, and when x=1, y=4, when x=2, y=8, and when x=3, y=18, express y in terms of x. 20. Given that the area of a circle varies as the square of its radius, and that the area of a circle is 154 square feet when the radius is 7 feet; find the area of a circle whose radius is 10 feet 6 inches. 21. The area of a circle varies as the square of its diameter; prove that the area of a circle whose diameter is 2 inches is equal to the sum of the areas of two circles whose diameters are 13, and 2 inches respectively. 22. The pressure of wind on a plane surface varies jointly as the area of the surface, and the square of the wind's velocity. The pressure on a square foot is 1 lb. when the wind is moving at the rate of 15 miles per hour; find the velocity of the wind when the pressure on a square yard is 16 lbs. 23. The value of a silver coin varies directly as the square of its diameter, while its thickness remains the same ; it also varies directly as its thickness while its diameter remains the same. Two silver coins have their diameters in the ratio of 4:3. Find the ratio of their thicknesses if the value of the first be four times that of the second. 24. The volume of a circular cylinder varies as the square of the radius of the base when the height is the same, and as the height when the base is the same. The volume is 88 cubic feet when the height is 7 feet, and the radius of the base is 2 feet; what will be the height of a cylinder on a base of radius 9 feet, when the volume is 396 cubic feet? 341. A SUCCESSION of quantities formed according to some fixed law is called a series. The separate quantities are called terms of the series. ARITHMETICAL PROGRESSION. 342. DEFINITION. Quantities are said to be in Arithmetical Progression when they increase or decrease by a common difference. Thus each of the following series forms an Arithmetical Progression : 3, 7, 11, 15,.. a, a+d, a +2d, a +3d, The common difference is found by subtracting any term of the series from that which follows it. In the first of the above examples the common difference is 4; in the second it is –6; in the third it is d. 343. If we examine the series a, a+d, a +2d, a +3d,... we notice that in any term the coefficient of d is always less by one than the number of the term in the series. Thus the 3rd term is a +2d; 20th term is a +19d; If n be the number of terms, and if I denote the last, or nth term, we have l=a+ (n-1)d. 344. To find the sum of a number of terms in Arithmetical Progression. Let a denote the first term, d the common difference, and n the number of terms. Also let 1 denote the last term, and s the required sum; then s=a+a+d)+(a+2d)+ ... +(1-21)+(1-1)+1; and, by writing the series in the reverse order, s=l+(1-d)+(1-2) + ... + (a +2d )+(a+d)+a. Adding together these two series, 28 = (a +1)+(a+1)+(a+l) + ... to n terms =n(a+l), s=} (a+2) (1); and l=a+(n-1)d .(2) .. s= ö{2a +(n-1)d} (3). 345. In the last article we have three useful formulæ (1), (2), (3); in each of these any one of the letters may denote the unknown quantity when the three others are known. .. S n Example 1. Find the 20th and 35th terms of the series 38, 36, 34,...... Here the common difference is 36–38, or -2. ... the 20th term =38+19(-2) =0; and the 35th term =38+34(-2) Example 2. Find the sum of the series 51, 64, 8, ...... to 17 terms. Here the common difference is 14 ; hence from (3) 17 11 The sum 2 x 2 2 (11+20) =263). Example 3. The first term of a series is 5, the last 45, and the sum 400: find the number of terms, and the common difference. If n be the number of terms, then from (1) n 400=315+45); whence n = = 16. If d be the common difference 45= the 16th term =5+15d; whence d=23. EXAMPLES XXXVI. a. 1. Find the 27th and 41st terms in the series 5, 11, 17,... 2. Find the 13th and 109th terms in the series 71, 70, 69,... 3. Find the 17th and 54th terms in the series 10, 111, 13,... 4. Find the 20th and 13th terms in the series -3, -2, - 1, ... 5. Find the 90th and 16th terms in the series –4, 2:5, 9,... 6. Find the 37th and 89th terms in the series – 2:8, 0, 2:8, ... Find the last term in the following series : 7. 5, 7, 9, ... to 20 terms. 8. 7, 3, - 1,... to 15 terms. 9. 13}, 9, 4.,... to 13 terms. 10. 6, 1.2, 1.8, ... to 12 terms. 11. 2:7, 3:4, 4:1,... to 11 terms. 12. X, 2x, 3x, ... to 25 terms. 13. a-d, atd, a+3d, ... to 30 terms. 14. 2a- b, 4a-36, 6a – 56,... to 40 terms. Find the last term and sum of the following series : 15. 14, 64, 114,... to 20 terms. 16. 1, 1.2, 1.4,... to 12 terms. 1 1 3 17. 9, 5, 1,... to 100 terms. 18. to 21 terms, 4' 4' 4" 19. 3}, 1, -- 1},... to 19 terms. 20. 64, 96, 128, ... to 16 terms. Find the sum of the following series : 21. 5, 9, 13, ... to 19 terms. 22. 12, 9, 6, ... to 23 terms. 23. 4, 51, 61, ... to 37 terms. 24. 103, 9, 73, .., to 94 terms. 25. –3, 1, 5, ... to 17 terms. 26. 10, 93, 9.1, ... to 21 terms. 27. 2, 3p, 5p, ... to p terms. 28. 3a, a, – a, ... to a terms. 29. a, 0, - a, ... to a terms. 30. - 31, -2, 9, ... to p terms. |