**2.13 Variation :**

Observe the below mentioned examples which we encounter in
our daily life:

Example 1: If 180 men in 6 days, working 10 hours daily, can dig a
trench of 60m long, 1 m wide and 1 m deep, how many days are needed for 100
men, working 8 hours a day, to dig a trench of 100m long, 1.5m wide and 1.2m
deep?

Example 2: The weight of a body varies inversely as the square of
its distance from the centre of the earth. If the radius of the earth is about
6380 KM, how much would a 80 KG man weigh 1600 KM
above the surface of the earth?

How can we find solutions to such problems?

Variation means change. We say things have changed with
time and we hear quite often that this was not the case with our time and
things have changed. We often hear that in our time the rice is used to cost few
Anas per KG,
where is it now costs in multiples of 10 Rs. Has the price of
rice increased directly with passage of time?. No. it has come down also. So raise in the price of rice or rise in the price of gold, petrol or
any item has not always increased with passage of time. So price raise
is not directly proportional to time. Does the height of a person with increase in age? Though height grows continuously in the initial stage, after certain
age it stops. So height does
is not directly vary with age.

**2.13.1** **Direct variation(Proportion)**

Is there
some thing which always increase with time? Yes, distance covered
by moving train or bus always increases with the time as per its speed. We also
know that

Distance travelled = Speed*time or d=st. If time travelled is less, then
distance covered is less. In this case we say distance is directly proportional
to time and we write dt and read this relation as: d is
directly proportional to t

Note that d/t = k- a constant (speed). Here k is a constant which does not change and is called 'constant of proportionality' where as d and t are
variables. Interest paid by a bank to a
depositor or interest charged on a loan is directly proportional to the amount
when term and rate of interest is fixed.

We know circumference of a circle = 2pr. As the radius of a circle increases or decreases,
circumference increases or decreases. Also note that Cr because C/r = 2p which is a constant. Similarly
Area of a circle = pr^{2}. Thus Ar and A/r^{2}= p. Here also area of a circle increases or decreases with increase or
decrease in the radius. To know the distance between 2 places, we use maps. The
distance between two places in a map is multiplied scale mentioned in the map
to get actual distances(Ex 1 cm = 10 Kms). Is weight directly proportional
to age? –NO.

**2.13 Problem 1 :** The distance through which a body
falls from rest varies as square of time it takes to fall that distance. It is
known that body falls 64 cm in 2 seconds. How far does that body fall in 6 seconds?

**Solution:**

Note that we can not solve as in
unitary method like

2sec >>> 64cm

_{} 6sec
>>> (64/2)*6= 192

This is not the solution.

It is given that dt^{2}^{}

_{} d/t^{2}=
k

_{}k = 64/4= 16

Now k = 16= d/6^{2}=d/36

_{}d= 16*36= 576

Thus the body falls in 576 cms in 6 seconds

**2.13.2** **Inverse variation(Proportion)**

Have you heard about growers throwing tomatoes on the road
when there is bumper crop? Why it is so? Note that:

·
When
there is too much of supply of vegetables or paddy, wheat etc, the price of
them crashes.

·
When
there is too much of supply of fuel in the international market the price of
Petrol and Diesel drops

Examples:

1.
When
more number of people are involved in a manual work, time taken to complete the
job decrease

2.
We
reach a state of weightlessness when we move far away from earth

In direct variation, when a value of a variable increases, the value of another dependant
variable also increases.

However, in the examples listed above when a value of a
variable increases, the value of another
dependant variable decreases. In such
cases, product dependent on dependent variables is constant.

In other words, If x and y are 2 variables then x1/y. We also
say x inversely
varies with y and accordingly xy=k a constant.

Like this we can also have x1/y^{2} , x1/y^{4}
, x1/_{} . . . . and then xy^{2}, xy^{4},
x_{} will be constant respectively.

**2.13 Problem 2 :** When a ball is thrown upwards,
the time, T seconds during which the ball remains in the air is directly
proportional to the square root of the height, h meters. reached.
We know T=4.47sec when h=25m.

(i)
Find
the formula for T in terms of h

(ii)
Find
T when h=50

(iii)
If
the ball is thrown upwards and remains in the air for 5 seconds, find the
height reached.

**Solution:**

T_{}

_{}T= k_{}

_{}4.47= 5k

_{}k = 0.894^{}

_{}T = 0.894 _{}

When h =50

T=
0.894*_{}= 0.894*7.07_{} 6.32

When T
=5

_{}= T/k = 5/0.894_{}5.60_{}31.36m

^{ }

**2.13.3
Joint Variation**

Can a variation
depend upon multiple variables? We know the formula for interest calculation which for
simple interest and compound interest is

SI =
PTR/100

and

CI = P(1+R/100)^{T}-P

What do we observe? we observe that Simple Interest is directly proportional to
Principal P, term T and rate of interest R and Compound interest is dependent
on those three variables.

We also know that weight of a person depends upon the
distance he is away from the centre of
earth.

**2.13.4 Work, People, Days, hours**

We know the work done is directly proportional to
number of men(M), number of days(D) and number of hours(H).

In other words,

WM, WD, WH

_{} WM*D*H or
M*D*H/W = constant

**2.13 Problem 3 :** If 36 men can build a wall of
140M long in 21 days, how many men are required to build a similar wall of
length 50M in 18 days?

**Solution:**

Here W_{1}= 140, M_{1}=36,
D_{1}=21 and given W_{2}= 50, D_{2}=18 'H' is number of
hours though not given is equal. we need to find M_{2} such that

M*D*H/W = constant

Hence 36*21*H/140 = M_{2}*18*H/50

On solving we get M_{2}=
15

^{ }

**2.13 Problem 4 :** Tap A Can fill a cistern in 8
hours and tap B can empty it in 12
hours. How long will it take to fill the cistern if both of them are
opened together

**Solution:**

Part filled in 1 hour = (1/8-1/12)= (3-2)/24= 1/24.

Time taken to fill the tank= 24
hours

**2.13 Problem 5 :** If 180 men in 6 days, working 10
hours daily, can dig a trench of 60m long, 1 m wide and 1 m deep, how many days
are needed for 100 men, working 8 hours a day, to dig a trench of 100m long,
1.5m wide and 1.2m deep?

**Solution:**

Here W_{1}= 60*1*1, M_{1}=180,
D_{1}=6 and H_{1} =10, and given W_{2}= 100*1.5*1.2, M_{2}=100,
and H_{2} =8,
we need to find D_{2} such that

M*D*H/W = constant

M_{1}*D_{1}*H_{1}/W_{1
}= M_{2}*D_{2}*H_{2}/W_{2}

Hence 180*6*10/60 = 100*D_{2}*8*/(100*1.5*1.2)

On solving we get D_{2} =
40.5

**2.13 Problem 6 :** The weight of a body varies
inversely as the square of its distance from the centre of the earth. If the
radius of the earth is about 6380 KM, how much would a
80 KG man weigh 1600 KM above the surface of the earth?

**Solution:**

Here W1/d^{2}

Weight of the person when he is on
earth = 80 KG. d_{1}_{ }=Radius of earth=6380
KM

Let W_{2 } be his weight when he is 1600 KM above
the earth's surface

_{} W_{1 }d_{1}^{2}=
W_{2 }d_{2}^{2}

80*6380^{2}= W_{2}*7980^{2}

On solving we get W_{2}= 51.14

**2.13 Problem 7 :** Suppose A alone can perform a
work in 5 days more than A and B working together. Suppose
further, B working alone can complete the same work in 20 days more than A and
B working together. Find out how much time it will take A and B both working together?

**Solution:**

Let **x**
be the time taken by A
and B both working together to perform the work.

It is given that A alone can complete the work in x+5 days

It is also given that B alone can
complete the work in x+20 days

From formula, The time taken by both
A and B working together is(**=x**)= (x+5)*(x+20)/{(x+5)+(x+20)}

_{} x= x^{2}+25x+100/2x+25

_{}2x^{2}+25x= x^{2}+25x+100

_{} x^{2}=100

_{} x=10

The time taken by both A and B, when they work
together to complete the work is 10 days

**Verification:**

Time taken by A alone to complete
the work = x+5 = 15 days

Time taken by B alone to complete the work = x+25 = 30 days

From formula, the time taken by
both A and B, when they work together= 30*15/45= 10

**2.13 Problem 8 :** It takes 1 hour to fill a tank by
a pump A. 1 hour and 40 minutes using a pump B. A third pump C takes average
time of pump A and B to fill up the tank. Suppose the pumps A and B are started
together and 2 pumps of capacity C are used to drain the water. Will the tank ever get
filled? If so in how much time it gets filled ?

**Solution:**

Time taken by pump A = 60 Minutes

Time taken by pump B = 100 Minutes

Time taken by pump C = 80 Minutes

Let t be the time taken to fill up the tank

By formula

1/t = 1/60+1/100- 2(1/80)

= 1/60+1/100-1/40= (10+6-15)/600 = 1/600

Time taken to fill up the tank =
600 minutes= 10 hours

Summary of learning

xy , x/y = k a
constant

x1/y. xy=k a
constant

WM*D*H or
M*D*H/W = k a constant