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SOLUTION The equation is a = k To solve for k: 0.9 = / (0.6) (2.7) (0.3)2 k = (0.9) (0.3)2 (0.6) (2.7) = 0.05 To solve for a: pintly as a = (0.05) (0.4) (0.7) _(0.05) (0.28) lar base. (0.2)2 = 0.35 (0.04) cm and lume of EXAMPLE 5. The current I varies directly as the electromotive force E 12 cm? and inversely as the resistance R. If a current of 30 amperes flows through a system with 16 ohms resistance and electromotive force of 120 volts, find the current that a 200-volt electromotive force will send through the system. SOLUTION RE I = 4E R R k.(120) 4(200) 30 = 16 16 (30) (16) = 50 amperes y varies k = 4 ntities. 120 where k Math Fy Probably the most famous equation that made a worldwide impact was Albert Einstein's E = mc. ", and The equation states that the total energy E (in joule) of a body is directly proportional to its mass m. Einstein found the constant of Albert Einstein nd the 1879-1955 proportionality to be c2, the square of the speed of light, or approximately 9x1016 m'/s?.5. The time rey directly as the length of the ditch and inversely as the number men working on the job. 6. The volume of a circular cone varies jointly as its height and square of the radius of its base. /. The pitch of a stretched vibrating string varies directly as the square of the tension and inversely as the length of the string. 8. The pressure of a gas varies jointly as its density and its absolute temperature. B. Mentally solve for k. 9. 13= k(39) (3) 10. 210 = k(3) (70) 11. 30 ~ R(15) 4 12. 150 = k(5) (30) 13. 72= k (22 ) (32 ) 14. 40= k (10 ) 32 Written math Solve each problem. 15. If z varies jointly as x and y, and z = 6 when x = 4 and y = 10, find z when x = 20 and y = 8. 16. A varies directly as B and inversely as C. If B = 4, C = 2, and A = 10, find A if B =15 and C = 25. 17. D varies as r and s, and inversely as t. If D = 12, r = 3, s = 20, and t = 5, find D when r = 15, s = 4, and t = 8. 160Since, k = -, the equation becomes y = -xz. 3 SOLUTION 40 40 The To solve for y, substitute x = 4 and z = 40 in the equation. To s 3. y = -XZ 40 V = - 40 - ( 4 ) ( 40 ) y =12 To so EXAMPLE 3. The volume of a right circular cylinder varies jointly as its height and the square of the radius of its circular base. If the volume of such cylinder whose height is 8 cm and EXAMPLE whose radius is 3 cm is 727 cm , what is the volume of the cylinder if the radius is 5 cm and the height is 12 cm? SOLUTION V = khr V = Thr 72TT = k(8) (3)2 V =1 (12) (5)2 SOLUTION 727 = k(8)(9) V = 1(12) (25) 721 = 72k V=300m cm T=k A variation is referred to as combined variation if one quantity varies directly or jointly as the other quantities and inversely as other quantities. COMBINED VARIATION If z varies jointly as x and y and inversely as t, then z = kxy where k is the constant of variation. EXAMPLE 4. The variable a varies directly as the product of b and c, and mad inversely as the square of d. E - If a = 0.9 when b = 0.6, c = 2.7, and d = 0.3, find the value of a when b = 0.4, c = 0.7, and d = 0.2. (in to iersely as the Joint and Combined the distance 3.4 lumination CHO Variations nversely as Learning Competencies ce between The learner will be able to: is 10 km, 1. illustrate situations that involve joint variations or combined variations; e between 2. translate into joint or combined variation statement a relationship between two quantities given (a) a table of values, (b) a mathematical equation, (c) a graph, and vice versa; and 3 . ired varies solve problems involving joint or combined variations. e volume volume is JOINT VARIATION Joint variation takes place when one quantity varies directly as the when s is product of two or more other quantities. If z varies jointly as x and y, then z = kxy, where k is the constant of variation. EXAMPLE 1. The area of a triangle varies jointly as the base and the height. Write the variation equation. product price is SOLUTION: nd for a In symbol, the given statement is A obh. The corresponding variation equation is A = kbh, where k is the constant of variation. EXAMPLE 2. If y varies jointly as x and z, and y. = 6 when x = 10 and Then, z = 8, find y when x = 4 and z = 40. SOLUTION The equation is y = kxz. To solve for k, substitute x = 10, z = 8, y = 6 in the equation. 6= k(10) (8) 6=80k 6 -= k 80 3 =k 40 15718. G varies directly as Wand the square of D, and inversely as L. If W is 3, D is 6, and L is 8 when G is 2700, find G when Wis 4, Dis 10, and L is 12. 19. The area of the curved surface of a right circular cylinder varies jointly as its radius and altitude. If the area of such a surface whose radius is 12 cm and whose altitude is 10 cm is 2407 cm, what is the area of a similar surface whose radius is 8 cm and whose altitude is 9 cm?. 20. The area of a rhombus varies jointly as the lengths of the two diagonals. If a rhombus whose diagonals are 10 cm and 8 cm long has an area of 40 cm?, find the area of the rhombus whose diagonals are 11 cm and 15 cm. 21. The volume of a cylinder varies directly as its height and the square of its radius. What is the effect on the volume if the height is doubled and the radius is halved? 22. The safe load for a horizontal beam supported at both ends varies jointly as the width and the square of its depth and inversely as the distance between the supports. If a 4 cm by 6 cm beam, 10 m long supports 5 kg when standing on the edge, what is the safe load if the beam is turned on its side? Math Challenge Solve each problem. 23. If z varies directly as the square of x and inversely as the square