TOPIC 4: RATES AND VARIATIONS ~ MATHEMATICS FORM 3
worked. If a man receives 1,000 shilling for two hours work, his rate of pay 1000 ÷ 2 = 500 shillings per hour. From the above example, we find out that
- What is his rate of pay?
- At this rate, how much would he receive for 20 hours work?
- At this rate, how long must he work to receive 30,000 shillings?
Relate quantities of the same kind
Example 2
A
student is growing plants she measures the rate at which two of them
are growing. Plant A grew 5cm in 10 days, and plant B grew 8cm in 12
days. Which plant is growing more quickly?
Convert Tanzanian currency into other currencies
countries have different currencies. Normally money is changed from one
currency to another using what is called a Rate of Exchange.
If at a certain time there are 1,100 shillings to each UK pound (£), to
go from £to shillings, multiply by 1,100, and to go from shillings to
£divide by 1,100.
Example 3
Suppose the current rate of change between the Tanzanian shillings and the Euro is 650 Tsh per Euro.
- A tourist changes 200 euros to Tsh. How much does he get?
- A business woman changes 2,080,000 Tsh to euros. How much does she get?
The Concept of Direct Variation
Explain the concept of direct variation
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.
Also the amount of maize you buy is directly proportional to the amount of money you spend.
Solve problems on direct variations
Example 4
Suppose a weight of 2kg gives an extension of 5cm.
Given
that y is proportional to x such that, when x = 40, y = 5. Find an
equation giving y in terms of x and use it to find (a) y when x = 15 (b)
x when y = 20.
Draw graphs of direct variation
Example 6
The linear equation graph at the right shows that as thexvalue increases, so does theyvalue increase for the coordinates that lie on this line.
The Concept of Inverse Variation
Explain the concept of inverse variation
some cases one quantity increase at the same rate as another decrease.
For example, if the first quantity is doubled, the second quantity is
halved.
In
this case the quantities vary inversely, or they are inversely
proportional. e.g. The number of men employed to dig a field is
inversely proportional to the time it takes. Also the time to travel a
journey is inversely proportional to the speed. We use the same symbol
(∝) for proportionality.
Solve problems on inverse variations
Example 7
If the volume is 0.8m3 when the pressure when the is 250kg/m3, find the formula giving the volume vm3 in terms of the pressure P kg/m2. What is the volume when the pressure is increased to 1,000kg/m2?
Given
that y is inversely proportional to x, such that x = 8 when y = 15.
Find the formula connecting x and y by expressing y in terms of x and
use it to find (a) y when x = 10, (b) x when y = 3
- Find the equation giving p in terms of q
- Find q when p = 0.5
- Find p when q = 160
Given
that y is inversely proportional to x such that when y = 6, x =7. Find
the equation connecting x and y by expressing x in terms of y and hence
find x when y = 36
number of workers needed to repair a road is inversely proportional to
the time taken. If 12 workers can finish the repair in 10 days, how long
will 30 workers take?
Graphs Relating Inverse Variations
Draw graphs relating inverse variations
a quantity is proportional to a power of another quantity. For example
the area A of a circle is proportional to the square of its radius r,
The mass of spheres of a certain metal is proportional to the cube of
their radii. A sphere of radius 10cm has mass 42kg. Find the formula
giving the mass m kg in terms of radius r cm. Find the radius of the
sphere with mass 5.25 kg.
that M is proportional to the square of N and when N = 0.3, M = 2.7.
Find the equation giving M in terms of N, and hence find the value of:
- M when N = 1.5
- N when M = 0.3
Use joint variation in solving problems
If a quantity varies as the product of two other quantities then it varies jointly with them. eg. If y = 3vu2, then y varies jointly with v and u2.
1. Suppose a mass of a gas with volume Vm3 is under pressure P kg/m2 and has absolute temperature T0.
The
volume of the gas varies jointly with its absolute temperature and
inversely with its pressure. At a temperature of 300 k and pressure of
80kg/m2, the volume is 0.5m3. Find the formula for the volume in terms of T and P.
TOPIC 4: RATES AND VARIATIONS ~ MATHEMATICS FORM 3
m
varies jointly with p and q such that when p = 12 and q = 5 then m= 15.
Find m in terms of p and q and hence find m when P = 3 and q = 28
- Find an equation giving M in terms of N.
- Find M when N = 4
- Find N when M = 5.
- Find an equation connecting P and Q expressing P in terms of Q.
- Find P when Q = 9
When a body is moving rapidly through the air, the air resistance R
newtons is proportional to the square of the velocity Vm/s, At a
velocity of 50m/s, the air resistance is 20N.
- Find R in terms of V
- Find the resistance at 100m/s.
- Find B in terms of A and C.
- Find B when A = 8 and C= 2
5.
The mass m kg of a solid wooden cylinder varies with the height h (m)
and with the square of the radius r (m). If v = 0.2 and h = 1.4, then M =
150. Find m in terms of h and r.
Example 13
1.
A cylinder has radius 3cm and volume 10cm3. If the radius of the base
is increased to 4cm without altering the height of the cylinder what
effect does this have on the volume?
A
pyramid has a square base. If the height decreases by 10% but the
volume remains constant, what must the side of the base increase by?
(i.e What increase in the side will of set the decrease in the height?).
Exercise 6
1. A box has a square base of side 5cm. The volume of the box is 56cm3. If the sides increase by 10%, without the height changing, what is the new volume of the box?
A water tank holds 1,000 liters, and is in the shape of cuboids. The
lengths of the sides of the base are enlarged by a scale factor of 1.4
without altering the height. What volume will the tank now hold?
4. The height of a cylinder is reduced by 20%. What percentage change is needed in the radius, if the volume remains constant?
