Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics §6.8 Model by Variation Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu

6.7 Review § Any QUESTIONS About Any QUESTIONS About HomeWork MTH 55 Review § Any QUESTIONS About §6.7 → Formulas and Applications of Rational Equations Any QUESTIONS About HomeWork §6.7 → HW-29

§6.8 Direct and Inverse Variation Equations of Direct Variation Problem Solving with Direction Variation Equations of Inverse Variation Problem Solving with Inverse Variation

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 → 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.

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

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

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

Example  Direct Variation cont. Graphing the Equation of Variation

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

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 Familarize and Translate: The problem states that we have DIRECT VARIATION between B and T. Thus an equation B = kT applies

Example  Bolt Production cont.1 Carry Out: Solve for k: Thus the Equation of Variation: If T = 5 hrs: Note that k is a RATE with UNITS

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, find the pressure at a depth of 40 feet. SOLN: This is a case of Direct Variation

Example  Fluid Statics

Example  Fluid Statics Use k = 5 lb/ft3 in the Direct Variation Equation to find the pressure at a depth of 40ft

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 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

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

Example  Barn Building It takes 6 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.

Example  Barn Building cont.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

Example  Barn Building cont.2 Translate: Since inverse variation applies use Carry Out: Find the Constant of Proportionality

Example  Barn Building cont.3 Carry Out: The Eqn of Variation When N = 35. Find T Chk: A check might be done by repeating the computations or by noting that (4.3)(35) and (6)(25) are both 150.

Example  Barn Building cont.4 STATE: if It takes 6 hours for 25 people to raise a barn, then it should take 35 people about 4.3 hours to build the same barn

To Solve Variation Problems Determine from the language of the problem whether direct or inverse variation applies. 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. Write the equation of variation and use it, as needed, to find unknown values.

Applications Tips ReDux The Most Important Part of Solving REAL WORLD (Applied Math) Problems Translating The Two Keys to the Translation Use the LET Statement to ASSIGN VARIABLES (Letters) to Unknown Quantities Analyze the RELATIONSHIP Among the Variables and Constraints (Constants)

Solving Variation Problems Write the equation with the constant of variation, k. Substitute the given values of the variables into the equation in Step 1 to find the value of the constant k. Rewrite the equation in Step 1 with the value k from Step 2 Use the equation from Step 3 to answer the question posed in the problem.

Other Variation Relations Some Additional Variation Eqns: y varies directly as the nth power of x if there is some positive constant k such that y varies inversely as the nth 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.

Combined Variation The Previous Variation Forms can be combined to create additional equations z varies directly as x and INdirectly as y if there is some positive constant k such that w varies jointly as x & y and inversely as z to the nth power if there is some positive constant k such that

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 Inverse variation to a Power →

Example  Luminance Solve for the Variation Constant, k, Using the KNOW values of I & D Use the value of k, and D = 9ft in the variation eqn to find E(9ft) State: At 9ft the Lamp produces a Luminance of 2 units

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 = kr3

Example  Sphere Volume Now use KNOWN data to solve for k Now Substitute k = 4π/3 into the Eqn of Variation

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 in3

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.

Example  Newton’s Law Write the Gravitation Law Symbolically SOLUTION: Let m1 and m2 be the masses of the two objects and r be the distance between them; a Diagram:

Example  Newton’s Law Next LET: F ≡ the gravitational force between the objects G ≡ the Constant of (Direct) Variation; a.k.a., the constant of proportionality Thus Newton Gravitation Law in Symbolic form

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 universally characterizes the intrinsic strength of the gravitational force. If the masses m1 and m2 are measured in kilograms, r is measured in meters, and the force F is measured in newtons, then the value of G:

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 RE = 6.38 x 106 meters Mass of the Earth ME = 5.98 x 1034 kg SOLUTION: By Newton’s 1st Law Force = (mass)·(acceleration) →

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:

Example  Newton’s Law Carry Out

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

Example  Kinetic Energy Write the Equation of Variation Next Solve for the Variation Constant, k, using the known data

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/m2

Example  Kinetic Energy Use k = 33.33 J/m2 to refine the Variation Equation Next use the E(v) eqn to find v for E = 76.8 kJ2

Example  Kinetic Energy The v for E = 76.8 kJ Thus when E = 76.8 kJ the velocity is 48 m/s

WhiteBoard Work Problems From §6.8 Exercise Set 33, 38 KINETIC and POTENTIAL Energy Balance

All Done for Today Heat Flows Hot→Cold

Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu Chabot Mathematics Appendix Bruce Mayer, PE Licensed Electrical & Mechanical Engineer BMayer@ChabotCollege.edu –

Graph y = |x| Make T-table