3.6 Variation Functions

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3.6 Variation Functions

• a.k.a. Proportion functions

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POD

• Simplify

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POD

• Simplify

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Direct variation

• General form y kx
• where k is the constant of proportionality
• Examples (what are the constants of
  proportionality?)
• C 2pr
• A pr2 (A varies directly as the square
  of r.)
• V (4/3)pr3 (How would you say this?)
• The Dance

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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

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Another way to put it

• Direct variation functions resemble power
  functions of the form y xn, where n gt 0.
• y 3x y (¼)x2 y x1/2
• Inverse variation functions resemble power
  functions of the form y xn, where n lt 0.
• y x-2 y 6.3x-1/2 y 4xn-3

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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.
• Well see this in two slides.

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The method to find the equation

1. Determine if the situation reflects direct or
   indirect variation.
2. Write the general formula.
3. Use given values to find k.
4. Use k to write the specific formula.
5. Use the specific formula to solve the problem.

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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.

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• Answer equations
• 1.
• 2.
• 3.

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Use it

• Hookes 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?

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Use it

• Hookes 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.

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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.

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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

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Use it

• Poiseuilles 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?

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Use it

• Poiseuilles 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.