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3.6 Variation 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

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3.6 Variation Functions

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  1. 3.6 Variation Functions a.k.a. Proportion functions

  2. POD Simplify

  3. POD Simplify

  4. 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

  5. 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

  6. 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

  7. 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.

  8. 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.

  9. 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.

  10. Answer equations: 1. 2. 3.

  11. 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?

  12. 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.

  13. 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.

  14. 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

  15. 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?

  16. 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.

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