2 Inverse variation - leafmaths.wikispaces.com · This chapter will show you ... Squares, square roots, cubes and cube roots of integers ... 2 Inverse variation

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This chapter will show you …

● how to solve problems where two variables are connected by arelationship that varies in direct or inverse proportion

What you should already know

● Squares, square roots, cubes and cube roots of integers

● How to substitute values into algebraic expressions

● How to solve simple algebraic equations

Quick check

1 Write down the value of each of the following.

a 52 b √«««81 c 33 d 3√«««64

2 Calculate the value of y if x = 4.

a y = 3x2 b y = 1√«x

1 Direct variation

2 Inversevariation

Direct proportion Inverse proportion

Edex_Higher Math_22.qxd 16/03/06 12:56 Page 519

© HarperCollinsPublishers Limited 2007

The term direct variation has the same meaning as direct proportion.

There is direct variation (or direct proportion) between two variables when one variable is a simplemultiple of the other. That is, their ratio is a constant.

For example:

1 kilogram = 2.2 pounds There is a multiplying factor of 2.2 between kilograms and pounds.

Area of a circle = πr2 There is a multiplying factor of π between the area of a circle and thesquare of its radius.

An examination question involving direct variation usually requires you first to find this multiplying factor(called the constant of proportionality), then to use it to solve a problem.

The symbol for variation or proportion is ∝.

So the statement “Pay is directly proportional to time” can be mathematically written as

Pay ∝ Time

which implies that

Pay = k × Time

where k is the constant of proportionality.

There are three steps to be followed when solving a question involving proportionality.

Step 1: set up the proportionality equation (you may have to define variables).

Step 2: use the given information to find the constant of proportionality.

Step 3: substitute the constant of proportionality in the original equation and use this to findunknown values.

520

Direct variation22.1

Key wordsconstant of

proportionality, kdirect proportiondirect variation

This section will introduce you to:● direct variation and show you how to work out

the constant of proportionality



In each case, first find k, the constant of proportionality, and then the formula connecting the variables.

T is directly proportional to M. If T = 20 when M = 4, find the following.

a T when M = 3 b M when T = 10

W is directly proportional to F. If W = 45 when F = 3, find the following.

a W when F = 5 b F when W = 90

Q varies directly with P. If Q = 100 when P = 2, find the following.

a Q when P = 3 b P when Q = 300

X varies directly with Y. If X = 17.5 when Y = 7, find the following.

a X when Y = 9 b Y when X = 30

CHAPTER 22: VARIATION

521

EXERCISE 22A

EXAMPLE 1

The cost of an article is directly proportional to the time spent making it. An article taking 6 hours to make costs £30. Find the following.

a the cost of an article that takes 5 hours to make

b the length of time it takes to make an article costing £40

Step 1: Let C be the cost of making an article and t the time it takes. We then have:

C ∝ t

⇒ C = kt

where k is the constant of proportionality.

Note that we can “replace” the proportionality sign ∝ with = k to obtain the proportionalityequation.

Step 2: Since C = £30 when t = 6 hours, then 30 = 6k

⇒ = k

⇒ k = 5

Step 3: So the formula is C = 5t

a When t = 5 hours C = 5 × 5 = 25

So the cost is £25.

b When C = £40 40 = 5 × t

⇒ = t ⇒ t = 8

So the making time is 8 hours.

405

306



The distance covered by a train is directly proportional to the time taken. The train travels 105 milesin 3 hours.

a What distance will the train cover in 5 hours?

b What time will it take for the train to cover 280 miles?

The cost of fuel delivered to your door is directly proportional to the weight received. When 250 kgis delivered, it costs £47.50.

a How much will it cost to have 350 kg delivered?

b How much would be delivered if the cost were £33.25?

The number of children who can play safely in a playground is directly proportional to the area ofthe playground. A playground with an area of 210 m2 is safe for 60 children.

a How many children can safely play in a playground of area 154 m2?

b A playgroup has 24 children. What is the smallest playground area in which they could safely play?

Direct proportions involving squares, cubes and square roots

The process is the same as for a linear direct variation, as the next example shows.


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EXAMPLE 2

The cost of a circular badge is directly proportional to the square of its radius. The cost of a badge with a radius of 2 cm is 68p.

a Find the cost of a badge of radius 2.4 cm.

b Find the radius of a badge costing £1.53.

Step 1: Let C be the cost and r the radius of a badge. Then

C ∝ r2

⇒ C = kr2 where k is the constant of proportionality.

Step 2: C = 68p when r = 2 cm. So

68 = 4k

⇒ = k ⇒ k = 17

Hence the formula is C = 17r2

a When r = 2.4 cm C = 17 × 2.42p = 97.92p

Rounding off gives the cost as 98p.

b When C = 153p 153 = 17r2

⇒ = 9 = r2

⇒ r = √««9 = 3

Hence, the radius is 3 cm.

15317

684



In each case, first find k, the constant of proportionality, and then the formula connecting the variables.

T is directly proportional to x2. If T = 36 when x = 3, find the following.

a T when x = 5 b x when T = 400

W is directly proportional to M2. If W = 12 when M = 2, find the following.

a W when M = 3 b M when W = 75

E varies directly with √««C . If E = 40 when C = 25, find the following.

a E when C = 49 b C when E = 10.4

X is directly proportional to √««Y . If X = 128 when Y = 16, find the following.

a X when Y = 36 b Y when X = 48

P is directly proportional to f 3. If P = 400 when f = 10, find the following.

a P when f = 4 b f when P = 50

The cost of serving tea and biscuits varies directly with the square root of the number of people atthe buffet. It costs £25 to serve tea and biscuits to 100 people.

a How much will it cost to serve tea and biscuits to 400 people?

b For a cost of £37.50, how many could be served tea and biscuits?

In an experiment, the temperature, in °C, varied directly with the square of the pressure, inatmospheres. The temperature was 20 °C when the pressure was 5 atm.

a What will the temperature be at 2 atm? b What will the pressure be at 80 °C?

The weight, in grams, of ball bearings varies directly with the cube of the radius measured inmillimetres. A ball bearing of radius 4 mm has a weight of 115.2 g.

a What will a ball bearing of radius 6 mm weigh?

b A ball bearing has a weight of 48.6 g. What is its radius?

The energy, in J, of a particle varies directly with the square of its speed in m/s. A particle moving at20 m/s has 50 J of energy.

a How much energy has a particle moving at 4 m/s?

b At what speed is a particle moving if it has 200 J of energy?

The cost, in £, of a trip varies directly with the square root of the number of miles travelled. Thecost of a 100-mile trip is £35.

a What is the cost of a 500-mile trip (to the nearest £1)?

b What is the distance of a trip costing £70?


523

EXERCISE 22B



There is inverse variation between two variables when one variable is directly proportional to thereciprocal of the other. That is, the product of the two variables is constant. So, as one variable increases,the other decreases.

For example, the faster you travel over a given distance, the less time it takes. So there is an inversevariation between speed and time. We say speed is inversely proportional to time.

S ∝ and so S =

which can be written as ST = k .

In each case, first find the formula connecting the variables.

T is inversely proportional to m. If T = 6 when m = 2, find the following.

a T when m = 4 b m when T = 4.8

W is inversely proportional to x. If W = 5 when x = 12, find the following.

a W when x = 3 b x when W = 10

kT

1T

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Inverse variation22.2

Key wordsconstant of

proportionality, kinverse proportion

This section will introduce you to:● inverse variation and show you how to work out

the constant of proportionality

EXAMPLE 3

M is inversely proportional to R. If M = 9 when R = 4, find the following.

a M when R = 2 b R when M = 3

Step 1: M ∝ ⇒ M = where k is the constant of proportionality.

Step 2: When M = 9 and R = 4, we get 9 =

⇒ 9 × 4 = k ⇒ k = 36

Step 3: So the formula is M =

a When R = 2, then M = = 18

b When M = 3, then 3 = ⇒ 3R = 36 ⇒ R = 1236R

362

36R

k4

kR

1R

EXERCISE 22C



Q varies inversely with (5 – t ). If Q = 8 when t = 3, find the following.

a Q when t = 10 b t when Q = 16

M varies inversely with t2. If M = 9 when t = 2, find the following.

a M when t = 3 b t when M = 1.44

W is inversely proportional to √««T . If W = 6 when T = 16, find the following.

a W when T = 25 b T when W = 2.4

The grant available to a section of society was inversely proportional to the number of peopleneeding the grant. When 30 people needed a grant, they received £60 each.

a What would the grant have been if 120 people had needed one?

b If the grant had been £50 each, how many people would have received it?

While doing underwater tests in one part of an ocean, a team of scientists noticed that thetemperature in °C was inversely proportional to the depth in kilometres. When the temperature was6 °C, the scientists were at a depth of 4 km.

a What would the temperature have been at a depth of 8 km?

b To what depth would they have had to go to find the temperature at 2 °C?

A new engine was being tested, but it had serious problems. The distance it went, in km, withoutbreaking down was inversely proportional to the square of its speed in m/s. When the speed was 12 m/s, the engine lasted 3 km.

a Find the distance covered before a breakdown, when the speed is 15 m/s.

b On one test, the engine broke down after 6.75 km. What was the speed?

In a balloon it was noticed that the pressure, in atmospheres, was inversely proportional to thesquare root of the height, in metres. When the balloon was at a height of 25 m, the pressure was1.44 atm.

a What was the pressure at a height of 9 m?

b What would the height have been if the pressure was 0.72 atm?

The amount of waste which a firm produces, measured in tonnes per hour, is inversely proportionalto the square root of the size of the filter beds, measured in m2. At the moment, the firm produces1.25 tonnes per hour of waste, with filter beds of size 0.16 m2.

a The filter beds used to be only 0.01 m2. How much waste did the firm produce then?

b How much waste could be produced if the filter beds were 0.75 m2?


525



y is proportional to √x. Complete the table.

The energy, E, of an object moving horizontally isdirectly proportional to the speed, v, of the object. When the speed is 10 m/s the energy is 40 000 Joules.

a Find an equation connecting E and v.

b Find the speed of the object when the energy is14 400 Joules.

y is inversely proportional to the cube root of x. When y = 8, x = 1–8 .

a Find an expression for y in terms of x,

b Calculate

i the value of y when x = 1–––125 ,

ii the value of x when y = 2.

The mass of a cube is directly proportional to the cubeof its side. A cube with a side of 4 cm has a mass of320 grams. Calculate the side length of a cube madeof the same material with a mass of 36 450 grams

y is directly proportional to the cube of x. When y = 16, x = 3. Find the value of y when x = 6.

d is directly proportional to the square of t. d = 80 when t = 4

a Express d in terms of t.b Work out the value of d when t = 7.

c Work out the positive value of t when d = 45.

Edexcel, Question 16, Paper 5 Higher, June 2005

The force, F, between two magnets is inverselyproportional to the square of the distance, x, betweenthem.

When x = 3, F = 4.

a Find an expression for F in terms of x.

b Calculate F when x = 2.

c Calculate x when F = 64.

Edexcel, Question 17, Paper 5 Higher, June 2003

Two variables, x and y, are known to be proportionalto each other. When x = 10, y = 25.

Find the constant of proportionality, k, if:

a y ∝ xb y ∝ x2

c y ∝ 1–x

d √««y ∝ 1–x

y is directly proportional to the cube root of x. When x = 27, y = 6.

a Find the value of y when x = 125.

b Find the value of x when y = 3.

The surface area, A, of a solid is directly proportionalto the square of the depth, d. When d = 6, A = 12π.

a Find the value of A when d = 12.Give your answer in terms of π.

b Find the value of d when A = 27π.

r is inversely proportional to t.r = 12 when t = 0.2

Calculate the value of r when t = 4.

Edexcel, Question 4, Paper 13B Higher, January 2003

The frequency, f, of sound is inversely proportional tothe wavelength, w. A sound with a frequency of 36 hertz has a wavelength of 20.25 metres.

Calculate the frequency when the frequency and thewavelength have the same numerical value.

t is proportional to m3.

a When m = 6, t = 324.Find the value of t when m = 10.

Also, m is inversely proportional to the square root of w.

b When t = 12, w = 25.Find the value of w when m = 4.

P and Q are positive quantities. P is inverselyproportional to Q2. When P = 160, Q = 20. Find the value of P when P = Q.

400

20

25

10

xy

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527

WORKED EXAM QUESTION

First set up the proportionality relationshipand replace the proportionality sign with = k.

Substitute the given values of y and x intothe proportionality equation to find thevalue of k.

Substitute the value of k to get the finalequation connecting y and x.

Substitute the value of x into the equationto find y.

Solution

M = kh3

4000 = k ×× 1000 ⇒ k = 4

So, M = 4h3

A = ph2

50 = p ×× 100 ⇒ p =

So, A = h2

32 000 = 4h3

h3 = 8000 ⇒ h = 20

A = (20)2 = = 200400

212

12

12

First, find the relationship between M andh using the given information.

Next, find the relationship between A and husing the given information.

Now find the value of A for that value of h.

Find the value of h when M = 32 000.

y is inversely proportional to the square of x. When y is 40, x = 5.

a Find an equation connecting x and y.

b Find the value of y when x = 10.

Solution

a y ∝∝

y =

40 =

⇒⇒ k = 40 ×× 25 = 1000

y =

or yx2 = 1000

b When x = 10, y = = = 101000100

1000102

1000x2

k25

kx2

1x2

The mass of a solid, M, is directly proportional to the cube of its height, h. When h = 10, M = 4000.

The surface area, A, of the solid is directly proportional to the square of the height, h. When h = 10, A = 50.

Find A, when M = 32 000.



An electricity company wants to build some offshore wind turbines (asshown below). The company is concerned about how big the turbines willlook to a person standing on the shore. It asks an engineer to calculatethe angle of elevation from the shore to the highest point of a turbine,when it is rotating, if the turbine was placed at different distances out tosea. Help the engineer to complete the first table below.

The power available in the wind ismeasured in watts per metre squaredof rotor area (W/m2). Wind speed ismeasured in metres per second (m/s).The power available in the wind isproportional to the cube of its speed.A wind speed of 7 m/s can provide210 W/m2 of energy. Complete thetable below to show the availablepower at different wind speeds.

Distance of Angle of elevationturbine out to sea from shore

3km 2.29º

4km

5km

6km

7km

8km

Wind speed Available power (m/s) (W/m2)

6 7 210 8 9 10 11 12

70m

50m

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Variation

The engineer investigates the different amountsof power produced by different length rotorblades at different wind speeds. He calculatesthe rotor area for each blade length – this is thearea of the circle made by the rotors – and thenworks out the power produced by these bladesat the different wind speeds shown. Help him tocomplete the table.

Remember

1 W = 1 watt1000 W = 1 kW = 1 kilowatt

1000 kW = 1 MW = 1 megawatt

Wind speed Available power Rotor area for Power Rotor area for Power Rotor area for Power (m/s) (W/m2) 50 m blade (m2) (MW) 60 m blade (m2) (MW) 70 m blade (m2) (MW)

7 210 7854 1.65

8 7854

9 7854

10 7854

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530

GRADE YOURSELF

Able to find formulae describing direct or inverse variation and usethem to solve problems

Able to solve direct and inverse variation problems involving threevariables

What you should know now

● How to recognise direct and inverse variation

● What a constant of proportionality is, and how to find it

● How to find formulae describing inverse or direct variation

● How to solve problems involving direct or inverse variation



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2 Inverse variation - leafmaths.wikispaces.com · This chapter will show you ... Squares, square roots, cubes and cube roots of integers ... 2 Inverse variation