Variations

The Concept of Direct Variation

Explain the concept of direct variation Some quantities are connected in such a way that they increase and decrease together at the same rate. Afar example if one quantity is doubled the other quantity is also doubled. These quantities are Directly Proportional or Vary Directly, Eg, Ifa ear is driven at a constant speed, the distance it goes is direetly proportional to the time taken Also the amount of maize you buy is directly proportional to the amount of money you spend.

Ify is directty proportional to x, we write YO X. This expression is not an equation. To change it to an equation, introduce a constant (k) called the constant of proportionality So y & X, then y=kx for some constant k.

Themes

Problems on Direct Variations

Solve problems on direct variations

Example 4

• Suppose different weights are hung from a wire, The extension of the wite is proportional to

the weight hanging. Suppose a weight of 2kg gives an extension of Sem Find an equation giving the extension © em in terms of weight w kg. Find the weight for an extension of Sem n

Solution

From the statement above, ¢ cw Soe=kw.ork=~

qetieis apg alte

‘A weight of L2kg gives an extension of Sem

Examples

Solution

yoke eZ

st weaatx et a

• Find an equation siving m in terms of a
• Find m when a= 18

• The mass m ke of apiece of metal is proportional to its volume Vim®. The mass of

0.2m is 210 ke (@) Find the equation connecting m and (®) Find the volume of I4ke of the metal (© Draw the graph of m agains v

Graphs of Direct Variation

Draw graphs of direct variation

Example 6

The linear equation graph at the right shows that as thervalue increases, 50 does theyvalue imerease forthe coordinates that lie om this line ye “4 we a 7. 10 x af aff abd at/

Cs A Toate Ab

fa This i a graph ofdirect variation ‘The Concept of Inverse Variation Explain the concept of inverse variation In some cases one quantity increase a the same rate as another decrease. For example, ifthe firs ‘quantity is doubled, the sccond quantity is halved n In this ease the quantities vary inversely, or they are inversely proportional. eg. The number of men employed to dig a field is inversely proportional to the time it takes. Also the time to travel

a joumey is inversely proportional to the speed We use the same symbol («) for proportionality

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Problems on Inverse Variations

Solve problems on inverse variations

Example 7

• Suppose a mass of a gas is kept at a constant temperature, The volume of the gas is inversely

proportional to its pressure If the volume is 0.8m' when the pressure when the is 250kg/m', find the formula giving the volume vm’ in terms of the pressure P kg/m?. What is the volume when the pressure is increased to 1,000kg/m”?

Solution

a k w+, soy=*or sp= “oD por Spek

P

P 1000

«The volume is 0.2m!

Example 8

good:

• x when y=3

Graphs Relating Inverse Variations

P Draw graphs relating inverse variations

Y A > 4 This is an example of a reciprocal graph Proportion to powers: Sometimes a quantity is proportional to a power of another quantity, For example the area A of a cirele is proportional to the square of its radius r,

‘The constant of the above equation is TT whose value is 22 or 3142 7

‘Therefore A= TE

Sometimes a quantity is inversely proportional to a power of another quantity. E.g. Newton's law of gravity states that the force of attraction (F) between two bodies is inversely proportional to L

Example 9

a2

• The mass of spheres of a certain metal is proportional to the cube of their radii. A sphere of

radius 10em has mass 42kg, Find the formula giving the mass m kg in terms of radius r em. Find the radius ofthe sphere with mass $.25 ke i mee Som=kr* To find k, put the values of m and r in the equation.

P2525

• The radius ofthe sphere is Sem

Example 10

giving M in terms of Nand hence find the value of:

b Nwhen M=03

Solution

Mean? M=kN? k=-M Fc

When M=2:7.N=03

o3

• M=? when N = 15

• N=? When M= 03

N30

N=VO0T=01

Joint Variation in Solving Problems. Use joint variation in solving problems Ia quantity varies as the product of wo other quantities then it varies jointly with them. eg. 'y

Example 11

• Suppose @ mass of a gas with volume Vm’ is under pressure P kg/m? and has absolute

temperature T° The volume ofthe gas varies jointly with its absolute temperature and inversely with its pressure ‘Ata temperature of 300 k and pressure of 80kg/m?, the volume is 0.5m" Find the formula for the volume in terms of T and P.

a

Solution

v the product of T and >, varies as the product of T and 5, SoV=I P \P-k z

geese “is kr 27 Venese PisP ar West 15P

Example 12

8s

Then m 2 2 Find an equation giving M in terms of N.

P

• B varies jointly with A and the inverse of C. When A= 3 and C= 12 then B = 20.

a Find B in terms of A and C

• The mass m kg ofa solid wooden cylinder varies with the height h (m) and with the square of

Joint variation leading to areas and volumes Many formulas for areas and volumes involve joint variation. For example the volume of

So the volume varies jointly with the height and the square of the radius. Le veoh.

Example 13

• A cylinder has radius 3em and volume 10cm3. If the radius of the base is increased to dem

without altering the height of the cylinder what effect does this have on the volume?

Solution

‘Where srs a constant and the height is constant, so v is proportional to the squate of +

Alternatively, as voc,

[z is constant, ifthe new volume is v. then 35— 75

Example 14

A pyramid has a square base. IF the height decreases by 10% but the volume remains constant what must the side ofthe base increase by? (i.e What inerease in the side will of set the decrease in the height?)

Solution

‘y= Fh. where isthe side of the base and his the height. ‘The volume varies jointly with band the square of. ‘Verhr?, so 7° is constant Let the height change from hi to hzand r from rt, to. while v remains constant v v so = haeay? —h2(r2)?

So if h changes from hy to ho, thea hy =am hi(ri)? 2102)? 1 10 n_ fe

(eH? 972)? mV?

: The side of the base must increase by 5.4%

• A box has a square base of side Sem. ‘The volume of the box is S6cm’. If the sides increase by

10%, without the height changing, what is the new volume of the box?

• A cone has volume 30cm’ If the radius increases by 10% and the height by 5%, what i the

new volume of the cone?

• A water tank holds 1,000 liters, and is in the shape of cuboids, The lengths ofthe sides of the

base are enlarged by a scale factor of 14 without altering the height, What volume will the tank ow hold?

• The height of'a cylinder is reduced by 20%, What percentage change is needed in the radius, if

the volume remains constant?