Joint variations & Part variations Lam Bo Jun (9) Lam Bo Xiang (10) Leong Zhao Hong (11)

Recap: Direct Variation Indirect Variation

Steps to do Variations Questions Write the equation Substitute given values Find the constant of variation, k Solve the values

Physics equations are included to facilitate revision for tomorrow’s test 

Joint variations

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.

Joint variations

Joint variations Distance Speed Time

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?

Joint variations (Example) From the question, Sub. in the values given: y = 10 when x = 5 and a = 3

Joint variations (Example) Now that we know the k = 6, we sub. in the value given in the question again: x = 13, a = 5

Some real life examples… The volume of a cylinder V = π r2 h The power of a heater (in Watts) Specific Heat Capacity

Part variations

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

Part variations Fixed profit Selling Price Initial price of good

Part variations

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

Part variations (Example) From the question,

Part variations (Example) Now that we know that the constant is 2, we sub. the values from the question in:

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]

Practice exercises

CYLINDER Variations q1

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/π)cm

Variations (Question 1) From question, V = k r2 h Therefore k = π

PRESSURE Variations q2

Variations (Question 2) Pressure varies directly with temperature and indirectly with volume. The internal pressure of a tank is 500Pa when the temperature is 40°C and its volume is 50cm3. What is its internal pressure when the temperature drops to 30°C and its volume is increased to 70cm3?

Variations (Question 2) From the question, Sub. in the values given:

Variations (Question 2) Now that we know the constant is 625, we sub. in the value given in the question again: Pa (3s.f.)

PRICE http://news-libraries.mit.edu/blog/wp-content/uploads/2008/01/money.jpg Variations q3

Variations (Question 3) The selling price of a certain good set by the shop owner is directly correlated to the good’s 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. What’s the initial price of a telephone if it’s sold at $190?

Variations (Question 3) From the question,

Variations (Question 3) Now that we know that the constant is 1.5, we sub. the values of the question in:

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

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 S$40 How happy will the customers be when the diameter of the pizza becomes 30cm but price rises to S$60 (for the same flavour of pizza) due to inflation?

Variations (Question 4) From the question:

Variations (Question 4) Now we know that k = 20, Sub D and P to get U

VELOCITY VARIATIONS Q5

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

Variations (Question 5) From question,

Variations (Question 5) Now we know that a = 0.1 m/s After another 20s, u = 2.5 m/s. Therefore:

THRUST VARIATIONS Q6

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%

Variations (Question 6)

MACHINES VARIATIONS Q7

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?

Variations (Question 7)

Bibliography http://www.regentsprep.org/Regents/math/algtrig/ATE7/Inverse%20Variation.htm http://www.regentsprep.org/Regents/math/algtrig/ATE7/variation%20practice%202.htm http://www.onlinemathlearning.com/joint-variation.html http://www.hci.sg/~angcc/Sec3Online/independentstudies.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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