Variation PDF

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P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> C H A P T E R 4 Variation E Objectives PL To recognise relationships involving direct variation To evaluate the constant of variation in cases involving direct variation To solve problems involving direct variation To recognise relationships involving inverse variation To evaluate the constant of variation in cases involving inverse variation To solve problems involving inverse variation To establish the relationship that exists between variables from given data To recognise relationships involving joint variation To solve problems involving joint variation M To solve problems involving part variation 4.1 Direct variation Emily sets out to drive from her home in Appleton to visit her friend Kim who lives 600 km away in Brownsville. She drives at a constant speed and notes how far she has travelled every SA hour. The distance and times are represented in the table below. Time (t hours) 1 2 3 4 5 6 Distance (d km) 100 200 300 400 500 600 It can be seen that as t increases, d also increases. The rule relating time to distance is d = 100t. This is an example of direct variation and 100 is the constant of variation. In this case d varies directly as t or the distance travelled is proportional to the time spent travelling. The graph of d against t is a straight line passing through the origin. A metal ball is dropped from the top of a tall building and the distance it has fallen is recorded each second. Time (t s) 0 1 2 3 4 5 Distance (d km) 0 4.91 19.64 44.19 78.56 122.75 89 Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> 90 Essential Advanced General Mathematics It can be seen that as t increases, d also increases. This time the rule relating time and distance is d = 4.91t 2. This is another example of direct variation. In this case, d varies directly as the square of t or the distance travelled is proportional to t 2. The graph of d against t 2 is a straight line passing through the origin. The symbol used for ‘varies as’ or ‘is proportional to’ is ∝. For example, d varies as t can be written as d ∝ t, and d varies as t 2 can be written as d ∝ t 2. In the following, a proportional to a positive power of b is considered, i.e. a varies directly as bn , n ∈ R + If a ∝ bn then a = kbn where k is a constant of variation. E For all examples of direct variation (where k is positive), as one variable increases the other will also increase. The graph of a against b will show an upwards trend. It should be noted that not all increasing trends will be examples of direct variation. If a ∝ bn then the graph of a against bn is a straight line passing through the origin. Example 1 a y ∝ x2 x y 2 12 4 PL Use the tables of values below to determine the constant of variation, k, in each case and hence complete each of the tables. 6 108 192 √
b y ∝ x i.e. y ∝ x 2 x y 1 2 4 1 6 1.225 1.414 M Solution √ a If y ∝ x2 b If y∝ x √ then y = kx 2 then y =k x When x = 2, y = 12 When x = 4, y = 1 √ ∴ 12 = k(22 ) ∴ 1 = k( 4) k =3 k = 0.5 SA Check: Check: √ When x = 6, y = 3(6 ) 2 When x = 6, y = 0.5( 6) = 108 ≈ 1.225 √ ∴ y = 3x 2 ∴ y = 0.5 x In order to complete the In order to complete the table, table, consider the following. consider the following. When x = 4, y = 3(42 ) √ When x = 2, y = 0.5( 2) y = 48 y ≈ 0.7071 When y = 192, 192 = 3x 2 √ When y = 1.414, 1.414 ≈ 0.5( x) 64 = x 2 √ 2.828 ≈ x x=8 x ≈8 Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> Chapter 4 — Variation 91 x 2 4 6 8 x 2 4 6 8 y 12 48 108 192 y 0.707 1 1.225 1.414 Example 2 In an electrical wire, the resistance (R ohms) varies directly with the length (L m) of the wire. a If a wire 6 m long has a resistance of 5 ohms, what would be the resistance in a wire of length 4.5 m? E b How long is a wire for which the resistance is 3.8 ohms? Solution The constant of variation is determined first. R∝L ∴ ∴ PL When L = 6, R = 5 R = kL 5 = k(6) k= 5 6 i.e. the constant of variation is Hence R= 5L 6 5 6 5 × 4.5 5L M a When L = 4.5, R = b When R = 3.8, 3.8 = 6 6 R = 3.75 L = 4.56 The resistance of a wire of length The length of a wire of resistance 4.5 m is 3.75 ohms. 3.8 ohms is 4.56 m. Example 3 SA The volume of a sphere varies directly as the cube of its radius. By what percentage will the volume increase if the radius is a doubled b increased by 20%? Solution V ∝ r3 i.e. V = kr 3 Initially set the radius equal to 1, then V = k(13 ) = k a If r is doubled, then set r = 2 Then V = k(23 ) = 8k Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> 92 Essential Advanced General Mathematics ∴ the volume has increased from k to 8k, an increase of 7k 7k 100 ∴ % increase of volume = × k 1 = 700% b If r is increased by 20%, then set r = 1.2. Then V = k(1.23 ) = 1.728k ∴ % increase of volume = 72.8% E Exercise 4A Example 1 1 Determine the value of k, the constant of variation, in each of the following and hence complete the table of values. a y ∝ x2 c y∝ x y x √ 2 8 x 4 PL 4 32 9 6 49 128 b y∝x d y ∝ x5 x y 1 1 2 1 6 1 1 3 2 1 2 2 3 M x 1 32 y 6 9 90 32 1 2 8 y 5 5 5 2 If V ∝ r 3 and V = 125 when r = 2.5, find a V when r = 3.2 b r when V = 200 SA 2 2 3 If a ∝ b3 and a = when b = 1, find 3 a a when b = 2 b b when a = 2 Example 2 4 The area (A) of a triangle of fixed base length varies directly as its perpendicular height (h). If the area of the triangle is 60 cm2 when its height is 10 cm, find a the area when its height is 12 cm b the height when its area is 120 cm2. 5 The extension in a spring (E) varies directly with the weight (w) suspended from it. If a weight of 452 g produces an extension of 3.2 cm, find a the extension produced by a weight of 810 g b the weight that would produce an extension of 10 cm. 6 The weight (W) of a square sheet of lead varies directly with the square of its side length (L). If a sheet of side length 20 cm weighs 18 kg, find the weight of a sheet that has an area of 225 cm2. Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> Chapter 4 — Variation 93 7 The volume (V) of a sphere varies directly with the cube of its radius (r). A sphere whose radius is 10 cm has a volume of 4188.8 cm3. Find the radius of a sphere whose volume is 1 cubic metre. Example 3 8 The time taken for one complete oscillation of a pendulum is called its period. The period (T) of a pendulum varies directly with the square root of the length (L) of the pendulum. A pendulum of length 60 cm has a period of 1.55 seconds. Find the period of a pendulum that is one and a half times as long. E 9 The distance (d) to the visible horizon varies directly with the square root of the height (h) of the observer above sea level. An observer 1.8 m tall can see 4.8 km out to sea when standing on the shoreline. a How far could the person see if they climbed a 4 m tower? b If the top of a 10 m mast on a yacht is just visible to the observer in the tower, how far 4.2 a doubled i y ∝ x2 Inverse variation PL out to sea is the yacht? 10 In each of the following calculate the percentage change in y when x is d increased by 40% b halved ii y ∝ √ x c reduced by 20% iii y ∝ x 3 A builder employs a number of bricklayers to build a brick wall. Three bricklayers will M complete the wall in eight hours but if he employs six bricklayers the wall will be complete in half the time. The more bricklayers he employs, the shorter the time taken to complete the wall. The time taken (t) decreases as the number of bricklayers (b) increases. This is an example of inverse variation. The time taken to complete the wall varies inversely as the number of bricklayers employed. t varies inversely as b or t is inversely proportional to b SA 1 i.e. t∝ b 1 In general, inverse variation exists if a ∝ where n is some positive number bn i.e. a varies inversely as bn. 1 If a∝ n b k then a = n where k is a positive constant called the constant of variation. b For all examples of inverse variation, as one variable increases the other will decrease and vice versa. The graph of a against b will show a downward trend. It should be noted, however, that any graph showing a decreasing trend will not necessarily be an example of inverse variation. 1 1 If a ∝ n then the graph of a against n will be a straight line. b b 1 However, since if b = 0, n is undefined, the line will not be defined at the origin. b Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> 94 Essential Advanced General Mathematics Example 4 Use the tables of values below to determine the value of the constant of variation, k, in each case and hence complete each of the tables. 1 1 a y∝ 2 b y∝√ x x x 2 5 10 x 1 25 100 y 0.1 0.016 0.001 y 10 5 1 E Solution 1 1 a y∝ b y∝√ x2 x k k ∴ y= 2 ∴ y=√ ∴ Check: When PL When x = 2, y = 0.1 0.1 = 2 x 2 k k = 0.4 i.e. the constant of variation is 0.4 x = 5, y = 0.4 52 = 0.16 When ∴ Check: When x = 1, y = 10 10 = √ k k = 10 1 x x = 100, y = √ =1 10 100 M 0.4 ∴ y= 2 10 x ∴ y=√ x In order to complete the table, consider the following. In order to complete the table, consider the following. 0.4 When x = 10, y = 10 102 When x = 4, y = √ y = 0.004 SA 4 0.4 y =5 When y = 0.001, 0.001 = 2 10 x When y = 2, 2 = √ 0.001x 2 = 0.4 x √ 0.4 2 x = 10 x2 = 0.001 x = 25 ∴ x = 20 x 2 5 10 20 x 1 4 25 100 y 0.1 0.016 0.004 0.001 y 10 5 2 1 Example 5 For a cylinder of fixed volume, the height (h cm) is inversely proportional to the square of the radius (r cm). Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> Chapter 4 — Variation 95 a What percentage change in the height would result if its radius were reduced by 25%? b If a cylinder 15 cm high has a base radius of 4.2 cm, how high would a cylinder of equivalent volume be if its radius were 3.5 cm? Solution 1 k a h∝ b h= r2 r2 k When h = 15, r = 4.2 i.e. h= 2 k r ∴ 15 = k (4.2)2 If r = 1, then h= =k E (1)2 k = 15(4.2)2 If r is reduced by 25%, then = 264.6 264.6 set r = 0.75 ∴ h= r2 k Then h= Consider a cylinder of radius 3.5 cm. (0.75)2 Exercise 4B PL = k 0.5625 ≈ 1.778k (correct to three decimal places) ∴ h is increased by 77.8% If r = 3.5, then h = h = 21.6 264.6 (3.5)2 The height of the cylinder is 21.6 cm. M Example 4 1 Determine the value of k, the constant of variation, in each of the following and hence complete the tables of values. 1 1 a y∝ b y∝√ x x x 2 4 6 1 x 1 9 SA 1 1 4 y 1 2 16 1 1 y 1 2 4 1 1 c y∝ d y∝ 1 x2 x3 x 1 2 3 1 3 1 x 1 125 y 3 8 4 12 2 1 1 y 3 3 9 1 √ 2 If a ∝ 3 and a = 4 when b = 2, find b √ 1 a a when b = 2 2 b b when a = 16 Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> 96 Essential Advanced General Mathematics 1 3 If a ∝ and a = 5 when b = 2, find b4 a a when b = 4 b b when a = 20. 4 The gas in a cylindrical canister occupies a volume of 22.5 cm3 and exerts a pressure of 1.9 kg/cm2. If the volume (V) varies inversely as the pressure (P), find the pressure if the volume is reduced to 15 cm3. Example 5 5 The current (I amperes) that flows in an electrical appliance varies inversely as the resistance (R ohms) of the appliance. If the current is 3 amperes when the resistance is E 80 ohms, find a the current when the resistance is 100 ohms b the increase in resistance required to reduce the current to 80% of its original value. 6 The intensity of illumination (I) of a light varies inversely as the square of the distance (d) PL from the light. At a distance of 20 m a light has an intensity of 100 candela. Find the intensity of the light at a distance of 25 m. 7 The radius (r) of a cylinder of fixed volume varies inversely as the square root of its height (h). If the height is 10 cm when the radius is 5.64 cm, find the radius if the height is 12 cm. 8 In each of the following, calculate the percentage change in y when x is a doubled d increased by 40%. b halved c reduced by 20% M 1 1 1 i y∝ 2 ii y ∝ √ iii y ∝ x x x3 4.3 Fitting data Sometimes the relationship that exists between two variables a and b is not known. By inspection of a table of values, it is sometimes possible to ascertain whether the relationship SA between the variables is direct or inverse proportion. Analysis is required to establish the rule that best fits the given data. This may involve graphing the data. Example 6 Establish the relationship between the two variables for each of the following tables of values. a b 0 2 4 6 8 b x 1 3 6 12 15 a 0 12 48 108 192 y 30 10 5 2.5 2 Solution a By inspection it can be conjectured that some type of direct variation exists. As b increases, a also increases and when a = 0, b = 0. Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> Chapter 4 — Variation 97 Assume a ∝ bn for some positive number n ∴ a = kbn a i.e. k= n b Select a value for n (it must be a positive number) and test each of the pairs of values given in the table (do not use (0, 0)). If the value of k for each pair of values is the same then the choice of n is correct. a a Let n = 1 ∴ k = Let n = 2 ∴ k = 2 b b E a a Consider for the values Consider 2 for the values b b given in the table. given in the table. 12 12 Testing: =6 Testing: =3 2 4 108 6 192 8 PL 48 4 = 12 = 18 = 24 Since the quotients differ, n = 1. 48 16 108 36 192 64 =3 =3 =3 The quotients are all equal to 3. ∴ k = 3 and n = 2 i.e. a = 3b2 M b By inspection it can be conjectured that some type of inverse variation exists. As x increases, y decreases. 1 Assume y∝ for some positive number n xn k ∴ y= n x SA i.e. k = yx n Let n = 1 ∴ k = yx Consider the product yx for the values given in the table. Testing: 30 × 1 = 30 10 × 3 = 30 5 × 6 = 30 2.5 × 12 = 30 2 × 15 = 30 ∴ k = 30 and n = 1 30 i.e. y= x The type of variation can also be investigated by graphical analysis. By plotting the graph of a against b, an upward trend may indicate direct variation or a downward trend may indicate inverse variation. Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> 98 Essential Advanced General Mathematics To find the specific type of variation that exists, the following can be used as a guide. If direct variation exists (a ∝ bn ), then the graph of a against bn will be a straight line through the origin. The gradient of this line will be the constant of variation k.   1 1 If inverse variation exists a ∝ n , then the graph of a against n will be a straight line. b b This line will not be defined at the origin. The gradient of this line will be the constant of variation k. Example 7 E For the table of values below, plot the graph of a against b2 and hence establish the rule relating a to b. b 1 2 3 4 5 a a Solution b2 0.5 1 0.5 PL4 2 2 4.5 9 4.5 8 16 8 12.5 25 12.5 Since this is a straight line, it can be conjectured that the relationship is a = kb2 where k corresponds to the 10 a 5 M 0 5 10 15 20 25 b2 gradient of the graph. 1 2 From the graph it can be seen that a = b. 2 If it is known that the relationship between two variables x and y is of the form y = kx n where k ∈ R + and n ∈ Q\{0} then a CAS calculator can be used to find n and k is SA sufficient information is given. Example 8 The following data was collected recording N, the number of calls to a company, D days after the commencement of an advertising campaign. Days (D) 5 10 15 Number of calls (N ) 50 400 1350 Find a relationship between N and D using the graphics calculator. Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> Chapter 4 — Variation 99 Solution Using the TI-Nspire Store the x-values and y-values as shown. E Select Power Regression from Stat Calculations submenu of the Statistics menu (b 6 1 ) and complete as shown. Press enter, and the result is PL given as y = a ∗ x ∧ b, a = 0.4, b = 3. Hence y = 0.4x 3 , so the required relationship is N = 0.4D 3. The graph and the data can be graphed in a Graphs & Geometry application ( 2) as a Function (b 3 1) and a Scatter Plot (b 3 4) respectively. M Using the Casio ClassPad In the program area, enter the data SA into list 1 and list 2 then tap Calc, Power Reg and ensure the settings are as shown. Note that selecting y1 will copy the formula to graph y1 in the program area. Note the formula from the Stat Calculation screen before tapping OK. The required relationship is N = 0.4D 3. Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> 100 Essential Advanced General Mathematics Example 9 Establish a rule connecting y and x given the following data. x 1 8 64 y 5 2.5 1.25 Solution The solution is given in the screens below. E Using the TI-Nspire PL Note thaty = 5x − 3 = 1 5 x3 1 M Using the Casio ClassPad 5 The solution is given in the screens. Note that y = 5x − 3 = 1 x−3 1 SA Exercise 4C Example 6 1 Each of the tables in parts a to e fits one of the following types of variation: 1 1 direct y ∝ x inverse y ∝ direct square y ∝ x 2 inverse square y ∝ 2 √ x x direct square root y ∝ x Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> Chapter 4 — Variation 101 Establish the relationship between x and y in each case. x 0 3 6 9 12 x 1 2 3 4 5 a b y 0 2 4 6 8 y 4 16 36 64 100 x 20 15 10 5 1 x 1 2 3 4 5 c 1 1 1 d y 1 5 y 2 2.828 3.464 4 4.472 4 3 2 x 1 1.5 2 2.5 3 e E y 4 1.78 1 0.64 0.444 2 Which of the following graphs could represent examples of direct variation? a y b y c y d 0 y PL x x e 0 y x3 1 f 0 y x x M 0 0 x 0 3 Which of the following graphs could represent examples of inverse variation? a y b y c y SA 1 x x 0 0 x 0 d y e y f y 1 x x 0 0 x3 0 Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> 102 Essential Advanced General Mathematics 4 Give the rule connecting y and x for each of the following. a y b y c y (3, 10) (1, 3) (2, 6) x 1 x2 0 0 x 0 d y e y f y E (9, 3) (1, 6) (1, 2) 1 √x x3 0 0 √x 0 Example 7 x y 2 9.6 2.5 15 6 Plot the graph of y against x y 1 1.5 4 3 PL 5 Plot the graph of y against x2 and hence establish the relationship between x and y. 9 4.5 3 21.6 16 6 3.5 29.4 √ 25 7.5 4 38.4 x and hence establish the relationship between x and y. M 1 7 Plot the graph of y against and hence establish the relationship between x and y. x2 x 0.2 0.3 0.4 0.5 1 y 50 22.2 12.5 8 2 SA Example 8 8 Given that for each of the following y ∝ ax b use your graphics calculator’s PwrReg function to establish the values of a and b. a x 4.00 8.00 12.00 16.00 b x 1 5 10 15 y 0.50 0.71 0.87 1.00 y 2.00 14.95 35.57 59.04 c x 1 10 100 1000 d x 10 20 30 40 y 3.50 8.79 22.08 55.47 y 46.42 73.68 96.55 116.96 e x 1 2 3 4 f x 1 3 5 7 y 2.00 0.35 0.13 0.06 y 3.20 2.06 1.68 1.47 Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> Chapter 4 — Variation 103 Example 9 9 The concentration of antibodies (C) in an animal’s bloodstream is directly proportional to time (t hours) after which the animal is injected with an antigen (i.e., C = at b ). The following data is collected. t 1 2 3 4 C 100 114.87 124.57 131.95 a Find values for a and b. b Find the concentration after 10 hours. E 10 The level of infestation (I) of a pest in a crop is proportional to the time (t days) after which the crop is sprayed with an insecticide. The relationship can be modelled using the rule I = at b , t ≥ 1. The following data is collected. 4.4 t I 1 1500 2 1061 PL 3 866 a Find values for a and b. Joint variation 4 750 b Find the level of infestation after 10 days. There are many situations where one variable depends on more than one other variable. The variable is said to vary jointly as the other variables. For example, the volume of a cylinder M varies jointly as the square of the radius and the height. i.e. V ∝ r 2h or V = kr 2 h (the value of k is known to be
) Example 10 SA x2 Given that y ∝ , use this table of values to determine z x 2 4 10 the value of the constant of variation k and hence z 10 8 50 complete the table. y 2 2.5 4 Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> 104 Essential Advanced General Mathematics Solution x2 y∝ z kx 2 ∴ y= z When x = 2 and z = 10, y = 2 k(22 ) 2= 10 ∴ k =5 5x 2 y= E i.e. z When x = 4, z = 8 5(42 ) ∴ y= 8 y = 10 When ∴ When ∴ PL z = 50, y = 2.5 2.5 = x =5 4= 5(x 2 ) 25 = x x = 10, y = 4 50 2 5(102 ) 4z = 500 z x z y 2 10 2 4 8 10 5 50 2.5 10 125 4 M z = 125 Example 11 The speed (s) of a conveyor belt varies jointly as the diameter (d) of the cog around which it SA passes and the number of revolutions per second (n) the cog makes. The speed of a belt that passes round a cog of diameter 0.3 m, revolving 20 times per second, is 18.85 m/s. Find the value of a the constant of variation b the speed of a belt passing around a cog half as big revolving 30 times per second. Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> Chapter 4 — Variation 105 Solution a s ∝ dn i.e. s = kdn When n = 20 and d = 0.3, s = 18.85 18.85 = k(0.3)(20) ∴ k = 3.142 (correct to three decimal places) ∴ s = 3.142dn b When d = 0.15 and n = 30 E s = 3.142 (0.15) (30) s = 14.14 m/s (correct to two decimal places) Exercise 4D Example 10 1 Given that y ∝ x z PL , use this table of values to determine the value of the constant of variation k and hence complete the table. 2 Given that y ∝ xz, use this table of values to determine the value of the constant of variation k and hence complete the table. x z y x z y 2 10 1 2 10 10 4 2 10 4 8 60 0.5 50 25 10 15 10 4 M z 3 Given that y ∝ , use this table of values to x2 x 2 3 10 determine the value of the constant of variation z 10 4 50 k and hence complete the table. 15 4 y 6 4 SA 2 3 4 a varies directly as b2 and inversely as c. If a = 0.54 when b = 1.2 and c = 2, find a when b = 2.6 and c = 3.5. 5 z varies as the square root of x and inversely as the cube of y. If z = 1.46 when x = 5 and y = 1.5, find z when x = 4.8 and y = 2.3. Example 11 6 The simple interest (I) earned on an investment varies jointly as the interest rate (r) and the time (t) for which it is invested. If a sum of money invested at 6.5% per annum for two years earns $130, how much interest would the same amount of money earn if it were invested at 5.8% for three years? 7 The kinetic energy (E) of an object varies directly as its mass (m) and the square of its velocity (v). If the kinetic energy of an object with a mass of 2.5 kg moving at 15 m/s is 281.25 joules, find the energy of an object with a mass of 1.8 kg moving at 20 m/s. Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> 106 Essential Advanced General Mathematics 8 The resistance (R) in an electrical wire varies directly as its length (l) and inversely as the square of its diameter (d). Find the percentage change in R if a l is increased by 50% and d is reduced by 50% b l is decreased by 50% and d is increased by 50%. 9 The weight (W) that can be supported by a wooden beam varies directly as the square of its diameter (d) and inversely as its length (l). a What percentage increase in the diameter would be necessary for a beam twice as long to support an equivalent weight? E b What percentage change in the weight would be capable of being supported by a beam three times as long with twice the diameter? 10 If p varies as the square of q and inversely as the square root of r, what is the effect on p if a both q and r are doubled b q is doubled and r is halved? PL 11 a The tension in a spring (T) varies directly with the extension (x) and inversely with the natural length (l) of the spring. Compare the tension in a spring with a natural length of 3 m that is extended by 1 m with the tension in a second spring with a natural length of 2.7 m that is extended by 0.9 m. b The work done (W) in stretching a spring varies directly with the square of the extension (x) and inversely with the natural length of the spring (l). Compare the work done on the two springs in part a. M 4.5 Part variation The total cost ($C) of printing cards is made up of a fixed overhead charge ($b) plus an amount that varies directly as the number printed (n). i.e. C = b + kn The total surface area (A) of a closed cylinder of fixed height is made up of two parts. The SA area of the curved surface (2
r h), which varies as the radius, and the area of the two ends (2
r 2 ), which varies as the square of the radius. i.e. A = k1r + k2r 2 where k1 = 2
h and k2 = 2
are the constants of variation. These are examples of part variation. Part variation exists when the value of one variable is the sum of two or more quantities each of which varies independently in some way. In some cases, as in the first example above, one of those quantities may be constant. Example 12 A monthly telephone account (A) is made up of a fixed charge (c) for rental and servicing plus an amount that is proportional to the number of calls made (n). In January, 220 calls were made and the account was for $98.20. In February, 310 calls were made and the account was for $120.70. Find the fixed charge and the cost per call. Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> Chapter 4 — Variation 107 Solution A = c + kn, where c equals the fixed charge and k equals cost per call 98.20 = c + 220k... 1 120.70 = c + 310k... 2 Solving simultaneously, subtract 1 from 2 22.5 = 90k k = 0.25 E Substitute in 1 98.20 = c + 220 (0.25) = c + 55 c = 43.2 PL The fixed charge is $43.20 and the cost per call is $0.25, i.e. 25 cents. Example 13 The stopping distance of a tram (d) (i.e. the distance travelled by the tram after its brakes are applied) varies partly with the speed of the tram (s) and partly with the square of its speed. A tram travelling at 15 km/h can stop in 57 m and at 20 km/h in 96 m. Find the formula that relates s to d and hence the stopping distance of a tram travelling at 18 km/h. M Solution d = k1 s + k2 s 2 57 = 15k1 + 225k2... 1 96 = 20k1 + 400k2... 2 Multiply 1 by 4 and 2 by 3 SA 228 = 60k1 + 900k2... 3 288 = 60k1 + 1200k2... 4 Subtract 3 from 4 60 = 300k2 1 k2 = 5 Substitute in 1   1 57 = 15k1 + 225 5 57 = 15k1 + 45 12 k1 = 15 Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> 108 Essential Advanced General Mathematics 4 k1 = 5 4 1 ∴ d = s + s2 5 5 When s = 18 4 1 d = (18) + (18)2 5 5 d = 79.2 The stopping distance of the tram will be 79.2 m. E Exercise 4E Example 12 1 The cost of a taxi ride (C) is partly constant (b) and partly varies with the distance travelled (d). A ride of 22 km costs $42.40 and a ride of 25 km costs $47.80. Find the cost of a Example 13 journey of 17 km. PL 2 The cost of a wedding reception at Hillview Reception Centre includes a fixed overhead charge and an amount per guest. a If a reception for 50 people costs $2625 and a reception for 70 people costs $3575, find the fixed overhead charge and the cost per guest. b Hence find the total cost of a reception for 100 guests. 3 p is the sum of two numbers, one of which varies as x and the other, as the square of y. If p = 14 when x = 3 and y = 4, and p = 14.5 when x = 5 and y = 3, find p when x = 4 M and y = 5. 4 The cost of running a ferris wheel in an amusement park varies partly as the number of people who ride it and partly as the inverse of the number of people who ride it. If the running cost is $32 when 200 people ride it and $61 if 400 people ride it, find the running cost on a day when 360 people ride it. SA 5 The distance travelled (s) by a particle varies partly with time and partly with the square of time. If it travels 142.5 m in 3 s and 262.5 m in 5 s, find a how far it would travel in 6 s b how far it would travel in the sixth second. 6 The time taken (t) to load boxes onto a truck varies partly with the number of boxes (b) and partly with the inverse of the number of men (m) loading the boxes. If it takes one man 45 minutes to load ten boxes and two men 30 minutes to load eight boxes, how long would it take four men to load sixteen boxes? Cambridge University Press Uncorrected Sample Pages 978-0-521-61252-4 2008 © Evans, Lipson, Jones, Avery, TI-Nspire & Casio ClassPad material prepared in collaboration with Jan Honnens & David Hibbard P1: FXS/ABE P2: FXS 9780521740494c04.xml CUAU033-EVANS September 8, 2008 16:5 Back to Menu >>> Chapter 4 — Variation 109 Review Chapter summary Direct variation a ∝ bn , i.e. a varies directly as bn (n ∈ R + ) This implies a = kbn where k is the constant of variation (k ∈ R + ). As b increases, a will also increase. If a ∝ bn , the graph of a against bn is a straight line through the origin. E Inverse variation 1 a ∝ n , i.e. a varies inversely as bn (n ∈ R + ) b k This implies a = n where k is the constant of variation (k ∈ R + ). b 1 PL As b increases, a will decrease. 1 If a ∝ n , the graph of a against n is a straight line but is undefined at the origin. b Joint variation b One quantity varies with more than one other variable. This may