if the area of a circle increases at a uniform rate then prove that the parameter varies inversely as the radius
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fferentiating w.r.t t
t
on both sides we get,
dAdt=π.2r.drdt
d
A
d
t
=
π
.2
r
.
d
r
d
t
It is given that the area is increasing at a uniform rate.
∴dAdt=k⇒2πr.drdt=k
∴
d
A
d
t
=
k
⇒
2
π
r
.
d
r
d
t
=
k
Step 2
where k
k
is a constant ∴drdt=k2πr
∴
d
r
d
t
=
k
2
π
r
Perimeter of the circle is
p=2πr
p
=
2
π
r
Differentiatily w.r.t t
t
we get
dpdt=2π.drdt
d
p
d
t
=
2
π
.
d
r
d
t
Substituting for drdt
d
r
d
t
we get
dpdt=2π.k2πr
d
p
d
t
=
2
π
.
k
2
π
r
dpdt=kr
d
p
d
t
=
k
r
⇒dpdtα1r
⇒
d
p
d
t
α
1
r
Hence this proves that the perimeter varies inversely as the radius.
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