a.k.a. Proportion functions

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Presentation transcript:

a.k.a. Proportion functions 3.6 Variation Functions a.k.a. Proportion functions

POD Simplify

POD Simplify

Direct variation General form: y = kx where k is the constant of proportionality Examples (what are the constants of proportionality?) C = 2πr A = πr2 (“A varies directly as the square of r.”) V = (4/3)πr3 (How would you say this?) The Dance

Indirect variation General form: y = k/x (where k is what?) Example: I = 110/R where I is current, R is resistance, and 110 is in volts. What is the constant of proportionality? The Dance

Another way to put it Direct variation functions resemble power functions of the form y = xn, where n > 0. y = 3x y = (¼)x2 y = x1/2 Inverse variation functions resemble power functions of the form y = xn, where n < 0. y = x-2 y = 6.3x-1/2 y = 4xn-3

Multiple variables Often, variation is a combination of more than two variables. In this case, there is still a constant of proportionality, and the different variables fall in a numerator or denominator. We’ll see this in two slides.

The method to find the equation Determine if the situation reflects direct or indirect variation. Write the general formula. Use given values to find k. Use k to write the specific formula. Use the specific formula to solve the problem.

Use it Write the specific formula for each of the following: 1. u is directly proportional to v. If v = 30, then u = 12. 2. r varies directly as s and inversely as t. If s = -2, and t = 4, then r = 7. 3. y is directly proportional to the square root of x, and inversely proportional to the cube of z. If x = 9, and z = 2, then y = 5.

Answer equations: 1. 2. 3.

Use it Hooke’s Law states that the force F required to stretch a spring x units beyond its natural length is directly proportional to x. A weight of four pounds stretches a certain spring from its natural length of 10 inches to a length of 10.3 inches. Find the specific formula. What weight will stretch this spring to a length of 11.5 inches?

Use it Hooke’s Law states that the force F required to stretch a spring x units beyond its natural length is directly proportional to x. A weight of four pounds stretches a certain spring from its natural length of 10 inches to a length of 10.3 inches. Find the specific formula. F = (40/3)x What weight will stretch this spring to a length of 11.5 inches? F = (40/3)(1.5) = 20 lbs.

Use it The electrical resistance R of a wire varies directly as its length l and inversely as the square of its diameter d. A wire 100 feet long, having a diameter of 0.01 inches has a resistance of 25 ohms. Find the specific formula. Find the resistance of a wire made of the same material that has a diameter of 0.015 inches and is 50 feet long.

Use it The electrical resistance R of a wire varies directly as its length l and inversely as the square of its diameter d. A wire 100 feet long, having a diameter of 0.01 inches has a resistance of 25 ohms. Find the specific formula. R = .000025l/(d2) Find the resistance of a wire made of the same material that has a diameter of 0.015 inches and is 50 feet long. R = 50/9 ohms

Use it Poiseuille’s Law states that the blood flow rate F (in L/min) through a major artery is directly proportional to the product of the fourth power of the radius and the blood pressure P. Express F in terms of P, r, and k. During heavy exercise, normal blood flow rates sometimes triple. If the radius of a major artery increases by 10%, approximately how much harder must the heart pump if the flow rate triples?

Use it Poiseuille’s Law states that the blood flow rate F (in L/min) through a major artery is directly proportional to the product of the fourth power of the radius and the blood pressure P. Express F in terms of P, r, and k. F = kPr4 During heavy exercise, normal blood flow rates sometimes triple. If the radius of a major artery increases by 10%, approximately how much harder must the heart pump if the flow rate triples? About 2.05 times as much.