JOINT VARIATIONS

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JOINT VARIATIONSPART VARIATIONS

• Lam Bo Jun (9)Lam Bo Xiang (10)Leong Zhao
  Hong (11)

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Recap

• Direct Variation

• Indirect Variation

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Steps to do Variations Questions

1. Write the equation
2. Substitute given valuesFind the constant of
   variation, k
3. Solve the values

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• Physics equations are included to facilitate
  revision for tomorrows test
• ?

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Joint variations
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Joint variations

• Different combinations of direct and indirect
  variations
• Used when a certain variable is
• affected by two or more other variables,
• directly and/or indirectly.

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Joint variations
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Joint variations
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Joint variations (Example)

• y varies directly with x and indirectly with a.
• y 10 when x 5 and a 3
• What is its value of y when x 13 and a 5?

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Joint variations (Example)

• From the question,
• Sub. in the values given y 10 when x 5 and a
  3

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Joint variations (Example)

• Now that we know the k 6, we sub. in the value
  given in the question again x 13, a 5

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Some real life examples

• The volume of a cylinderV p r2 h
• The power of a heater (in Watts)
• Specific Heat Capacity

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Part variations
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Part variations

• Contains direct or indirect variation or even
  both
• Used when an extra constant is added to the
  affected variable, neither directly nor
  indirectly affecting it

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Part variations
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Part variations
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Part variations (Example)

• The value of y 5 is directly related to w and x
• When w 7 and x 3, y 37
• Find the value of y when w 13 and x 21

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Part variations (Example)

• From the question,

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Part variations (Example)

• Now that we know that the constant is 2, we sub.
  the values from the question in

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Some real life examples
And some PHYSICS revision ?

• Kinematics v u at v2 u2 2as s ut ½
  at2
• Dynamics F m (v u) / t

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Practice exercises
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VARIATIONS Q1

• CYLINDER

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Variations (Question 1)

• The volume of a cylinder is directly proportional
  to the square of the radius of the base and the
  height
• Find the value of the volume when the radius is
  7cm and the height is (30/p)cm

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Variations (Question 1)

• From question,V k r2 h
• Therefore k p

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VARIATIONS Q2

• PRESSURE

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Variations (Question 2)

• Pressure varies directly with temperature and
  indirectly with volume.
• The internal pressure of a tank is 500Pa when the
  temperature is 40C and its volume is 50cm3.
• What is its internal pressure when the
  temperature drops to 30C and its volume is
  increased to 70cm3?

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Variations (Question 2)

• From the question,
• Sub. in the values given

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Variations (Question 2)

• Now that we know the constant is 625, we sub. in
  the value given in the question again

Pa
(3s.f.)
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VARIATIONS Q3

• PRICE

http//news-libraries.mit.edu/blog/wp-content/uplo
ads/2008/01/money.jpg
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Variations (Question 3)

• The selling price of a certain good set by the
  shop owner is directly correlated to the goods
  initial buying price.
• On top of that, there is a fixed profit of 10
  for every item sold.
• The selling price of a book is 55 when its
  initial price is 30.
• Whats the initial price of a telephone if its
  sold at 190?

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Variations (Question 3)

• From the question,

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Variations (Question 3)

• Now that we know that the constant is 1.5, we
  sub. the values of the question in

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VARIATIONS Q4
http//www.resch-frisch.com/images/products/Pizza-
Capricciosa.jpg

• PIZZA

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Variations (Question 4)

• PizzaHub delights its customers with its cheap
  and delicious pizzas
• Suppose the average happiness of the customers
  (in utils) vary directly with the diameter of the
  pizza (in cm) and inversely with the price of the
  pizza (in S), and has a base of 5 utils
• The customers have an average of 30 utils when
  the pizza diameter is 50cm and the price of the
  pizza is S40
• How happy will the customers be when the diameter
  of the pizza becomes 30cm but price rises to S60
  (for the same flavour of pizza) due to inflation?

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Variations (Question 4)

• From the question

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Variations (Question 4)

• Now we know that k 20,Sub D and P to get U

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VARIATIONS Q5

• VELOCITY

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Variations (Question 5)

• A car travels at a constant velocity of 0.5 m/s
• It accelerates constantly to 2.5 m/s after 20s
• Given v u at, where v is the final velocity,
  u is the initial velocity, a is acceleration, t
  is time taken, find a
• Hence find the final velocity of the car after
  another 20s, given constant acceleration

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Variations (Question 5)

• From question,

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Variations (Question 5)

• Now we know that a 0.1 m/s
• After another 20s, u 2.5 m/s. Therefore

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VARIATIONS Q6

• THRUST

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Variations (Question 6)

• Under certain conditions, the thrust T of a
  propeller varies jointly as the fourth power of
  its diameter d and the square of the number n of
  revolutions per second
• Show that if n is doubled, and d is halved, the
  thrust T decreases by 75

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Variations (Question 6)
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VARIATIONS Q7

• MACHINES

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Variations (Question 7)

• The number of hours h that it takes m men to
  assemble x machines varies directly as the number
  of machines and inversely as the number of men.
• If four men can assemble 12 machines in four
  hours, how many men are needed to assemble 36
  machines in eight hours?

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Variations (Question 7)
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Bibliography

• http//www.regentsprep.org/Regents/math/algtrig/AT
  E7/Inverse20Variation.htm
• http//www.regentsprep.org/Regents/math/algtrig/AT
  E7/variation20practice202.htm
• http//www.onlinemathlearning.com/joint-variation.
  html
• http//www.hci.sg/angcc/Sec3Online/independentstu
  dies.html
• http//www.purplemath.com/modules/variatn.htm
• http//www.purplemath.com/modules/variatn2.htm
• http//www.purplemath.com/modules/variatn3.htm

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THANK YOU