1. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 1 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §6.8 Model by Variation

2. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 2 Bruce Mayer, PE Chabot College Mathematics Review §  Any QUESTIONS About §6.7 → Formulas and Applications of Rational Equations  Any QUESTIONS About HomeWork §6.7 → HW-28 6.7 MTH 55

3. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 3 Bruce Mayer, PE Chabot College Mathematics §6.8 Direct and Inverse Variation  Equations of Direct Variation  Problem Solving with Direction Variation  Equations of Inverse Variation  Problem Solving with Inverse Variation

4. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 4 Bruce Mayer, PE Chabot College Mathematics Direct Variation  Many problems lead to equations of the form y = kx, where k is a constant. Such eqns are called equations of variation  DIRECT VARIATION  DIRECT VARIATION → When a situation translates to an equation described by y = kx, with k a constant, we say that y varies directly as x. The equation y = kx is called an equation of direct variation.

5. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 5 Bruce Mayer, PE Chabot College Mathematics Variation Terminology  Note that for k > 0, any equation of the form y = kx indicates that as x increases, y increases as well  Synonyms “y varies as x,” “y is directly proportional to x,” “y is proportional to x”  The Synonym Terms also imply direct variation and are often used

6. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 6 Bruce Mayer, PE Chabot College Mathematics The Constant “k”  For the Direct Variation Equation  The constant k is called the constant of proportionality or the variation constant.  k can be found if one pair of values for x and y is known.  Once k is known, other (x,y) pairs can be determined

7. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 7 Bruce Mayer, PE Chabot College Mathematics Example  Direct Variation  If y varies directly as x, and y = 3 when x = 12, then find the eqn of variation  SOLUTION: The words “y varies directly as x” indicate an equation of the form y = kx:  Solving for k  Thus the Equation of Variation

8. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 8 Bruce Mayer, PE Chabot College Mathematics Example  Direct Variation cont.  Graphing the Equation of Variation  Direct Variation Always produces a SLANTED LINE that Passes Thru the ORIGIN

9. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 9 Bruce Mayer, PE Chabot College Mathematics Example  Direct Variation  Find an equation in which a varies directly as b, and a = 15 when b = 25.  Find the value of a when b = 36  SOLUTION:  Thus the Variation Eqn  Sub b = 36 into Eqn  Thus when b = 36, then the value of a is 21-3/5

10. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 10 Bruce Mayer, PE Chabot College Mathematics Example  Bolt Production  The number of bolts B that a machine can make varies directly as the time T that it operates.  The machine makes 3288 bolts in 2 hr  How many bolts can it make in 5 hr 1.Familarize and Translate: The problem states that we have DIRECT VARIATION between B and T. Thus an equation B = kT applies

11. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 11 Bruce Mayer, PE Chabot College Mathematics Example  Bolt Production cont.1 3.Carry Out: Solve for k: Thus the Equation of Variation: If T = 5 hrs: Note that k is a RATE with UNITS

12. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 12 Bruce Mayer, PE Chabot College Mathematics Example  Fluid Statics  The pressure exerted by a liquid at given point varies directly as the depth of the point beneath the surface of the liquid.  If a certain liquid exerts a pressure of 50 pounds per square foot (psf) at a depth of 10 feet, then find the pressure at a depth of 40 feet.  SOLN: Another case of Direct Variation

13. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 13 Bruce Mayer, PE Chabot College Mathematics Example  Fluid Statics (units are lb/ft 3 )

14. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 14 Bruce Mayer, PE Chabot College Mathematics Example  Fluid Statics  Use k = 5 lb/ft 3 in the Direct Variation Equation to find the pressure at a depth of 40ft

15. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 15 Bruce Mayer, PE Chabot College Mathematics Inverse Variation  INVERSE VARIATION  INVERSE VARIATION → When a situation translates to an equation described by y = k/x, with k a constant, we Say that y varies INVERSELY as x. The equation y = k/x k/x is called an equation of inverse variation Note that for k > 0, any equation of the form y = k/x indicates that as x increases, y decreases

16. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 16 Bruce Mayer, PE Chabot College Mathematics Example  Inverse Variation  If y varies inversely as x, and y = 30 when x = 20, find the eqn of variation  SOLUTION: The words “y varies inversely as x” indicate an equation of the form y = k/x:  Solving for k  Thus the Equation of Variation

17. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 17 Bruce Mayer, PE Chabot College Mathematics Example  Barn Building  It takes 56 hours for 25 people to raise a barn.  How long would it take 35 people to complete the job? Assume that all people are working at the same rate.

18. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 18 Bruce Mayer, PE Chabot College Mathematics Example  Barn Building cont.1 LET 1.Familarize. Think about the situation. What kind of variation applies? It seems reasonable that the greater number of people working on a job, the less time it will take. So LET: T ≡ the time to complete the job, in hours, N ≡ the number of people working  Then as N increases, T decreases and inverse variation applies

19. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 19 Bruce Mayer, PE Chabot College Mathematics Example  Barn Building cont.2 2.Translate: Since inverse variation applies use 3.Carry Out: Find the Constant of Inverse Proportionality

20. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 20 Bruce Mayer, PE Chabot College Mathematics Example  Barn Building cont.3 3.Carry Out: The Eqn of Variation  When N = 35. Find T 4.Chk: A check might be done by repeating the computations or by noting that (28.8)(35) and (56)(25) are both 1008.

21. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 21 Bruce Mayer, PE Chabot College Mathematics Example  Barn Building cont.4 5.STATE: if It takes 56 hours for 25 people to raise a barn, then it should take 35 people about 29 hours to build the same barn

22. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 22 Bruce Mayer, PE Chabot College Mathematics To Solve Variation Problems 1.Determine from the language of the problem whether direct or inverse variation applies. 2.Using an equation of the form y = kx for direct variation or y = k/x for inverse variation, substitute known values and solve for k. 3.Write the equation of variation and use it, as needed, to find unknown values.

23. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 23 Bruce Mayer, PE Chabot College Mathematics Applications Tips ReDux  The Most Important Part of Solving REAL WORLD (Applied Math) Problems  The Two Keys to the Translation LETUse the LET Statement to ASSIGN VARIABLES (Letters) to Unknown Quantities Analyze the RELATIONSHIP Among the Variables and Constraints (Constants)

24. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 24 Bruce Mayer, PE Chabot College Mathematics Solving Variation Problems 1.Write the equation with the constant of variation, k. 2.Substitute the given values of the variables into the equation in Step 1 to find the value of the constant k. 3.Rewrite the equation in Step 1 with the value k from Step 2 4.Use the equation from Step 3 to answer the question posed in the problem.

25. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 25 Bruce Mayer, PE Chabot College Mathematics Other Variation Relations  Some Additional Variation Eqns:  y varies directly as the n th power of x if there is some positive constant k such that  y varies inversely as the n th power of x if there is some positive constant k such that  y varies jointly as x and z if there is some positive constant k such that.

26. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 26 Bruce Mayer, PE Chabot College Mathematics Combined Variation  The Previous Variation Forms can be combined to create additional equations  z varies directly as x and INversely as y if there is some positive constant k such that  w varies jointly as x & y and inversely as z to the n th power if there is some positive constant k such that

27. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 27 Bruce Mayer, PE Chabot College Mathematics Example  Luminance  The Luminance of a light (E) varies directly with the intensity (I) of the light source and inversely with the square distance (D) from the light. At a distance of 10 feet, a light meter reads 3 units for a 50-cd lamp. Find the Luminance of a 27-cd lamp at a distance of 9 feet.  This is a case of COMBINED Variation →

28. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 28 Bruce Mayer, PE Chabot College Mathematics Example  Luminance  Solve for the Variation Constant, k, Using the KNOWN values of I & D  Use the value of k, and D = 9ft in the variation eqn to find E(9ft)  State: At 9ft the 27cd Lamp produces a Luminance of 2 cd/m 2

29. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 29 Bruce Mayer, PE Chabot College Mathematics Example  Sphere Volume  Suppose that you had forgotten the formula for the volume of a sphere, but were told that the volume V of a sphere varies directly as the cube of its radius r. In addition, you are given that V = 972π when r = 9in.  Find V when r = 6in  SOLUTION: Recognize as Direct Variation to a Power: V = kr 3

30. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 30 Bruce Mayer, PE Chabot College Mathematics Example  Sphere Volume  Now use KNOWN data to solve for k  Now Substitute k = 4π/3 into the Eqn of Variation

31. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 31 Bruce Mayer, PE Chabot College Mathematics Example  Sphere Volume  Finally Substitute r = 6 and solve for V(6) as requested  Using π ≈ 3.14159 find the Volume for a 6 inch radius sphere, V(6) ≈ 904.78 in 3

32. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 32 Bruce Mayer, PE Chabot College Mathematics Example  Newton’s Law  Newton’s Law of Universal Gravitation says that every object in the universe attracts every other object with a force acting along the line of the centers of the two objects and that this attracting force is directly proportional to the product of the two masses and inversely proportional to the square of the distance between the two objects.

33. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 33 Bruce Mayer, PE Chabot College Mathematics Example  Newton’s Law  Write the Gravitation Law Symbolically  SOLUTION: Let m 1 and m 2 be the masses of the two objects and r be the distance between them; a Diagram:

34. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 34 Bruce Mayer, PE Chabot College Mathematics Example  Newton’s Law  Next LET: F ≡ the gravitational force between the objects G ≡ the Constant of Variation; a.k.a., the constant of proportionality  Thus Newton Gravitation Law in Symbolic form

35. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 35 Bruce Mayer, PE Chabot College Mathematics Example  Newton’s Law  The constant of proportionality G is called the universal gravitational constant. It is termed a “universal constant” because it is thought to be the same at all places and all times and thus it universally characterizes the intrinsic strength of the gravitational force.  If the masses m 1 and m 2 are measured in kilograms, r is measured in meters, and the force F is measured in newtons, then the value of G:

36. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 36 Bruce Mayer, PE Chabot College Mathematics Example  Newton’s Law  Next Estimate the value of g; the “acceleration due to gravity” near the surface of the Earth. Use these estimates: Radius of Earth R E = 6.38 x 10 6 meters Mass of the Earth M E = 5.98 x 10 34 kg  SOLUTION: By Newton’s 1 st Law Force = (mass)·(acceleration) →

37. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 37 Bruce Mayer, PE Chabot College Mathematics Example  Newton’s Law  Now the “Force of Gravity” at the earth’s surface is the result of the Acceleration of Gravity:  Equating the “Force of Gravity” and the Gravitation Force Equations:

38. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 38 Bruce Mayer, PE Chabot College Mathematics Example  Newton’s Law  Carry Out

39. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 39 Bruce Mayer, PE Chabot College Mathematics Example  Kinetic Energy  The kinetic energy of an object varies directly as the square of its velocity.  If an object with a velocity of 24 meters per second has a kinetic energy of 19,200 joules, what is the velocity of an object with a kinetic energy of 76,800 joules?  SOLUTION: This is case of Direct Variation to the Power of 2

40. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 40 Bruce Mayer, PE Chabot College Mathematics Example  Kinetic Energy  Write the Equation of Variation  Next Solve for the Variation Constant, k, using the known data

41. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 41 Bruce Mayer, PE Chabot College Mathematics Example  Kinetic Energy  To find k, use the fact that an object with a velocity of 24 m/s has a kinetic energy of 19.2 kJ  Thus k = 33.33 J/m 2

42. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 42 Bruce Mayer, PE Chabot College Mathematics Example  Kinetic Energy  Use k = 33.33 J/m 2 to refine the Variation Equation  Next use the E(v) eqn to find v for E = 76.8 kJ 2

43. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 43 Bruce Mayer, PE Chabot College Mathematics Example  Kinetic Energy  The v for E = 76.8 kJ  Thus when E = 76.8 kJ the velocity is 48 m/s

44. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 44 Bruce Mayer, PE Chabot College Mathematics WhiteBoard Work  Problems From §6.8 Exercise Set 33, 38  KINETIC and POTENTIAL Energy Balance

45. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 45 Bruce Mayer, PE Chabot College Mathematics All Done for Today Heat Flows Hot → Cold

46. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 46 Bruce Mayer, PE Chabot College Mathematics Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix –

47. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 47 Bruce Mayer, PE Chabot College Mathematics Graph y = |x|  Make T-table

48. BMayer@ChabotCollege.edu MTH55_Lec-36_sec_6-8_Model_by_Variation.ppt 48 Bruce Mayer, PE Chabot College Mathematics