Question

7. If the area of a circle increases at a uniform
rate, then show that rate of change in perimeter
varies inversely as its radius. [NCERT]​

Answer 1

Given:

Area of a circle increases at a uniform

rate.

To Show:

Rate of change in perimeter varies inversely as its radius

Solution:

Area of circle = πr² =A

Let A be any constant

Differentiating with respect to t on both sides.

dA/dt = 2πr.dr/dt ...(1)

Now

Perimeter of circle = 2πr = P

Let P be any constant

Differentiating with respect to t on both sides.

dP/dt = 2π.dr/dt

dr/dt = 1/2π × dp/dt

Now putting value of dr/dt in eqn (1)

dA/dt = 2πr × 1/2π × dp/dt

dA/dt = r × dp/dt

dp/dt = 1/r × dA/dt

Hence, rate of change in perimeter varies inversely as its radius.

Answer 2

Step-by-step explanation:

Given : -

• .If the area of a circle increases at a uniform rate,.

To Find : -

• then show that rate of change in perimeter varies inversely as its radius

Solution : -

Area of circle = πr²

Differentiating w.r.t t on both sides we get,

dA/dt = 2πr × dr/dt

It is given that the area is increasing at uniform rate

therefore ,

dA/dt = K

2πr (dr/dt) =K

Where K is constant

therefore ,

dr/dt = K/2π r

Perimeter of the circle is p = 2πr

Differentiatily w.r.t t we get

dpdt = 2π × drdt

Substituting for drdt we get

dpdt = 2π × k2πr

dpdt = kr

dpdt = 1r

Hence this proves that the perimeter varies inversely as the radius.

.

More Information

Radius is a line from the center to the outside of a circle or sphere. .

• The definition of a radius is a circular limit or a boundary of a specific distance which is drawn from a specific point.