M110 CLASS NOTES

1
M110 CLASS NOTES

• SECTION 2.5
• Variation

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Types of Variation

• y varies _______ as x
• Directy y kx
• Inversely y k/x
• z varies jointly as x and y
• Joint z kxy
• Combined
• a combination of the above

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Types of Proportion

• y is _______ proportional to x
• Directy y kx
• Inversely y k/x
• z is jointly proportional to x and y
• Joint z kxy
• Combined
• a combination of the above

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EXAMPLE 1

• A is jointly proportional to B, C and the square
  of D. Express this statement as a formula if A
  is 144 when B is 10, C is 3 and D is 2.

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EXAMPLE 1

• A is jointly proportional to B, C and the square
  of D. Express this statement as a formula if A
  is 144 when B is 10, C is 3 and D is 2.

SOLUTION A kBCD2
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EXAMPLE 1

• A is jointly proportional to B, C and the square
  of D. Express this statement as a formula if A
  is 144 when B is 10, C is 3 and D is 2.

SOLUTION A kBCD2 144 k(10)(3)(2)2
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EXAMPLE 1

• A is jointly proportional to B, C and the square
  of D. Express this statement as a formula if A
  is 144 when B is 10, C is 3 and D is 2.

SOLUTION A kBCD2 144 k(10)(3)(2)2 144 120k
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EXAMPLE 1

• A is jointly proportional to B, C and the square
  of D. Express this statement as a formula if A
  is 144 when B is 10, C is 3 and D is 2.

SOLUTION A kBCD2 144 k(10)(3)(2)2 144
120k K 144/120
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EXAMPLE 1

• A is jointly proportional to B, C and the square
  of D. Express this statement as a formula if A
  is 144 when B is 10, C is 3 and D is 2.

SOLUTION A kBCD2 K 144/120 6/5
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EXAMPLE 1

• A is jointly proportional to B, C and the square
  of D. Express this statement as a formula if A
  is 144 when B is 10, C is 3 and D is 2.

SOLUTION A kBCD2 K 144/120 6/5 A
(6/5)BCD2
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EXAMPLE 2

• The volume, V, of a gas varies directly as its
  temperature, T, and inversely as its pressure, P.
  If 40 m3 of a gas yields a pressure of 20 kg/m
  at a temperature of 200 k, what will be the
  volume of the same gas if the pressure is
  decreased 10 kg/m and the temp. is increased to
  250 k?

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EXAMPLE 2

• The volume, V, of a gas varies directly as its
  temperature, T, and inversely as its pressure, P.
  If 40 m3 of a gas yields a pressure of 20 kg/m
  at a temperature of 200 k, what will be the
  volume of the same gas if the pressure is
  decreased 10 kg/m and the temp. is increased to
  250 k?

V kT/P
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EXAMPLE 2

• If 40 m3 of a gas yields a pressure of 20 kg/m
  at a temperature of 200 k, what will be the
  volume of the same gas if the pressure is
  decreased 10 kg/m and the temp. is increased to
  250 k?

V kT/P 40 k(200)/(20)
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EXAMPLE 2

• If 40 m3 of a gas yields a pressure of 20 kg/m
  at a temperature of 200 k, what will be the
  volume of the same gas if the pressure is
  decreased 10 kg/m and the temp. is increased to
  250 k?

V kT/P 40 k(200)/(20) 40 10k
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EXAMPLE 2

• If 40 m3 of a gas yields a pressure of 20 kg/m
  at a temperature of 200 k, what will be the
  volume of the same gas if the pressure is
  decreased 10 kg/m and the temp. is increased to
  250 k?

V kT/P 40 10k k 4
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EXAMPLE 2

• If 40 m3 of a gas yields a pressure of 20 kg/m
  at a temperature of 200 k, what will be the
  volume of the same gas if the pressure is
  decreased 10 kg/m and the temp. is increased to
  250 k?

V 4T/P
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EXAMPLE 2

• What will be the volume of the same gas if the
  pressure is decreased 10 kg/m and the temp. is
  increased to 250 k?

V 4T/P V 4(250)/(10)
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EXAMPLE 2

• What will be the volume of the same gas if the
  pressure is decreased 10 kg/m and the temp. is
  increased to 250 k?

V 4(250)/(10) V 100
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EXAMPLE 2

• What will be the volume of the same gas if the
  pressure is decreased 10 kg/m and the temp. is
  increased to 250 k?

V 100 m3
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EXAMPLE 3

• 32 page 197 The lift L on an airplane at
  takeoff varies jointly as the square of the speed
  s of the plane and the area A of its wings. A
  Plane with a wing area of 500 ft2 traveling at 50
  mi/hr experiences a lift of 1700 lb. How much
  lift would a plane with a wing area of 600 ft2
  traveling at 40 mi/hr experience?

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EXAMPLE 3

• The lift L on an airplane at takeoff varies
  jointly as the square of the speed s of the plane
  and the area A of its wings. A Plane with a wing
  area of 500 ft2 traveling at 50 mi/hr experiences
  a lift of 1700 lb. How much lift would a plane
  with a wing area of 600 ft2 traveling at 40 mi/hr
  experience?

L ks2A
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EXAMPLE 3

• A Plane with a wing area of 500 ft2 traveling at
  50 mi/hr experiences a lift of 1700 lb. How much
  lift would a plane with a wing area of 600 ft2
  traveling at 40 mi/hr experience?

L ks2A 1700 k(50)2(500)
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EXAMPLE 3

• A Plane with a wing area of 500 ft2 traveling at
  50 mi/hr experiences a lift of 1700 lb. How much
  lift would a plane with a wing area of 600 ft2
  traveling at 40 mi/hr experience?

L ks2A 1700 k(50)2(500) 1,250,000k
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EXAMPLE 3

• A Plane with a wing area of 500 ft2 traveling at
  50 mi/hr experiences a lift of 1700 lb. How much
  lift would a plane with a wing area of 600 ft2
  traveling at 40 mi/hr experience?

L ks2A 1700 1,250,000k k .00136
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EXAMPLE 3

• A Plane with a wing area of 500 ft2 traveling at
  50 mi/hr experiences a lift of 1700 lb. How much
  lift would a plane with a wing area of 600 ft2
  traveling at 40 mi/hr experience?

L .00136 s2A
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EXAMPLE 3

• How much lift would a plane with a wing area of
  600 ft2 traveling at 40 mi/hr experience?

L .00136 s2A L .00136(40)2(600)
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EXAMPLE 3

• How much lift would a plane with a wing area of
  600 ft2 traveling at 40 mi/hr experience?

L .00136 s2A L .00136(40)2(600) L 1305.6
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EXAMPLE 3

• How much lift would a plane with a wing area of
  600 ft2 traveling at 40 mi/hr experience?

L .00136 s2A L 1305.6 lb
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End of Section 2.5