(Q1).If the diameter of the cross – section of a wire is decreased by 5%, by what percentage should the length be increased so that the volume remains the same ?

Let the radius of wire = r

(if diameter = 2r)

and length of wire = h = l (say)

then volume of wire = πr2l1 = v1 ….(1)

Now the diameter after decreasing 5%

= 2r- 2r.
5
100
 =
200r - 10r
100
 =
r

then radius of wire after decreasing

=
190r
100
 ×
1r
2
 =
95r
100

and let the length of wire = (l2)

then volume of wire = π.(
95r
100
 )2 .l2 =v2 ….(2)

Both volume are same v1 = v1

⇒ l1 = .
.
. l2
⇒ l2 =
l1

Increase in l = l2 - l1

=
l1
 - l1
=
(10000 - 9025)l1
9025
=
975
9025
 l1 =
l1
% of increase in length =
(39/361)l1
l1
 ×100
=
39
l61
 ×100 = %

So it length is increased by 24.22%. Its volume remains same.