L1_Introduction to Variations_ Math 9_ Quarter 2.pptx

Math 9 lesson, quarter 2.

1. Introduction to Variations LESSON 1

2. REVIEW

3. 1. Direct Variation - exists whenever the ratio between two quantities is a nonzero constant. The statements: “y varies directly as x” “y is directly proportional to x” “y is proportional to x” …may be translated mathematically as y= kx , where k is the constant of variation. For two quantities, x and y, an increase in x causes an increase in y as well, a decrease in x causes a decrease in y.

4. Examples: “a varies directly as b” can be written as a = kb or k = “the cost (c) of egg varies directly as the number (n) of pieces bought” can be written as c = kn or k = “the distance (d) covered by a car is directly proportional to the time (t) it travels” can be written as d = kt or k =  

5. Activity 1 The amount (A) to pay in electric bill varies directly as the kilowatt-hours consumed (c). The electrical resistance (R) of a wire varies directly as its length (l). The number of teachers (t) needed in a school varies directly as the number of students (s) enrolled.

6. 2. Inverse Variation -occurs whenever a situation produces pairs of numbers whose product is constant. The statements: “y varies inversely to x” …may be translated mathematically as y= where k is the constant of variation. For two quantities, x and y, an increase in x causes a decrease in y or vice versa.  

7. Examples: “r varies inversely as s” can be written as r = or k= rs 2. “the numbers (n) of workers and the amount of time (t) to finish the work” can be written as n = or k= nt 3. “the force (F) of attraction between two opposite electrical charges varies inversely as the square of the distance (d) between them” can be written as F = or k= Fd 2  

8. Activity 2 t varies inversely as V . The number of pechay plants n in a row varies inversely as the space s between them. The length l of a rectangular field varies inversely as its width w . The base b of a triangle varies inversely as its altitude h .

9. Activity 3 The distance D an automobile can travel is directly proportional to the time t that it travels at a constant speed. The volume V of a given mass of gas is inversely proportional to the pressure p exerted on it An automobile’s braking distance d is directly proportional to the square of the automobile’s speed v The volume V of a sphere varies directly as the cube of its radius r The extension of a hanging spring d is directly proportional to the weight w attached to it.

10. Activity 3 6. The pressure P of a gas is inversely proportional to its volume V. 7. The volume V of a gas varies directly with the square of its temperature T. 8. The number of hours h it takes to complete a job varies inversely with the number of workers w. 9. The force F needed to break a board is inversely proportional to the board's length L. 10. The current I in a circuit varies inversely with the resistance R

11. Activity 3 The distance D an automobile can travel is directly proportional to the time t that it travels at a constant speed. D=kt or k= D/t The volume V of a given mass of gas is inversely proportional to the pressure p exerted on it. V=k/p or k= Vp An automobile’s braking distance d is directly proportional to the square of the automobile’s speed v. d=kv 2 or k= d/v 2 The volume V of a sphere varies directly as the cube of its radius r. V= kr 3 k= V/r 3 The extension of a hanging spring d is directly proportional to the weight w attached to it. d= kw or k=d/w

12. 6. The pressure P of a gas is inversely proportional to its volume V. P= k/V or k= PV 7. The volume V of a gas varies directly with the square of its temperature T. V= kT 2 or k=V/ T 2 8. The number of hours h it takes to complete a job varies inversely with the number of workers w. h= k/w or k= hw 9. The force F needed to break a board is inversely proportional to the board's length L. F=k/L or k= FL 10. The current I in a circuit varies inversely with the resistance R. I= k/R or k=IR

13. 3. Joint Variation - takes place when one quantity varies directly as the product of two or more quantities. The statements: “y varies jointly as x and z” …may be translated mathematically as y= kxz , where k is the constant of variation and x ≠ 0 and z ≠ 0.

14. Examples: “m varies jointly as n and p” can be written as m = knp “Area (A) of a rectangle varies jointly as its base (b) and height (h)” can be written as A = kbh “Volume (V) of a rectangular prism varies jointly as the base (b), height (h) and width (w)” can be written as V = kbhw

15. Activity 4 V varies jointly as l, w, and h. The volume of a cylinder V varies jointly as its height h and the square of the radius r. P varies jointly as q and r. W varies jointly as c and the square of a

16. 4. Combined Variation - occur when one quantity varies either directly or jointly as the other quantities and inversely as the other quantities. The statements: “y varies directly as x and inversely as z” …may be translated mathematically as y= , where k is the constant of variation and z ≠ 0.  

17. Examples: “e varies jointly as f and g and inversely as h” can be written as e = “the velocity (v) of a car varies directly with the distance (d) and inversely as the time (t)” can be written as v = “the time (t) it takes to construct a building varies directly as the height (h) of the building and inversely as the number (n) of workers” can be written as t=  

18. Activity 5 W varies jointly as c and the square of a and inversely as b. The pressure P of a gas varies directly as its temperature t and inversely as its volume V.

19. EXAMPLES

20. The fare F of a passenger varies directly as the distance d of his destination. The area A of a square varies directly as the square of its side s . The number of pechay plants n in a row varies inversely as the space s between them. The length l of a rectangular field varies inversely as its width w. The base b of a triangle varies inversely as its altitude h . V varies jointly as l , w , and h .

21. 7. The volume of a cylinder V varies jointly as its height h and the square of the radius r . 8. P varies jointly as q and r . 9. W varies jointly as c and the square of a and inversely as b . 10. The pressure P of a gas varies directly as its temperature t and inversely as its volume V .

22. V= 4s 3 F   N   V= 4s 3

23. V= 4s 3 F   N  

24. Directions: Express each of the following statements into an equation where k is the constant of variation. The amount (A) to pay in electric bill varies directly as the kilowatt-hours consumed (c). The number (n) of passengers in a jeep varies inversely as the space (s) between them. The electrical resistance (R) of a wire varies directly as its length (l) and inversely as the square of its diameter (d). The area (A) of a parallelogram varies jointly as the base (b) and height (h). The number of teachers (t) needed in a school varies directly as the number of students (s) enrolled.