The circumference of a circle increases from 20 pi to 30 pi by how many cm will the radius increase

Well, let's solve this first and discuss a bit more…

Firstly, Circumference of a circle = 2πr

Now it is increased by 20%.

Obviously, π cannot increase nor can 2.

So, radius is increased.

Since the increase is expressed in %, it now becomes relative.

Okay, what do I mean by relative?

Look how you obtain %..

%increase/decrease = 100 ∆r/r

So, its relative because it has a division by r thus relative to ‘r’.

So, what's great about being a relative measure?

You can assume it,the radius, to be any arbitrary value.. It doesn't matter.. After all, you are dividing it with at the end.

So, no harm in assuming r = 100.

Why exactly 100?

Look at the definition of ‘percentage’. It's a conversion formula which scales to a factor of 100. If the number is itself 100… there is no need to scale at all!!

Now, increase by 20% = increase by 20 w.r.t. 100

But our choice of r is itself 100.

Hence new r = r' = 120.

Now, new area = π(r')^2 = 14400π

Old area = 10000π.

Now %increase = 100∆A/A = 44%.

So, increase is 44%.

Well, now come a new question,

What if the %r is small, say x%

i.e., 100∆r/r = x%

Should I say the answer by solving in the above method and tell?

:D No!

Now see,

A(r) =πr^2

Take natural log on both sides…

ln[A(r)] = lnπ + 2lnr

Now differentiate on both sides..

1/A(r) [dA(r)/dr] = 1/r.

Then we have,

dA/A = dr/r

100dA/A =100dr/r

Now, if we assume x% change in r is so small then dr = ∆r and dA = ∆A

So, we have 100∆A/A = 100∆r/r = 2x%

So, what do we can say from above?

For small relative changes in radius, effect on area will be twice of that!

If we assume 20% to be small, then for your problem 40% is the answer by above approach against to the actual answer of 44%

Why is this useful?

Suppose you are creating a disc made of mud and thought of increasing the radius of the disc by 20%. You want to see how much more money you need to put to make that change. Assuming the mud cost is very low you don't bother if I say 40% roughly or 44% percent exactly. It justs make a difference of 4% which might be around 2INR.

But suppose you are creating a disc made of diamond and wanted to increase the radius by 20% and I told you that it would cost you roughly 40% more but to be precise 44%, you would yell at me. Because, 4% interns of diamond can be some 1000’s of INR (:D assume some 1mm thickness)

So, it's relative, the quantification of the term ‘small’

step-by-step explanation:

if x = 2 you have to find x to finish the problem