Question

The rate of change of perimeter of circle with respect to its radius at r =5 cm

Answer 1

Please see the attachment

Answer 2

Given,

The perimeter of a circle.

To find,

Rate of change of perimeter of a circle with respect to its radius, at r = 5 cm.

Solution,

The perimeter of a circle, that is the distance around the circle, is called the circumference, and is given by,

$C=2\pi r$     ...(1)

Where,

C is the circumference (perimeter) of the circle, and

r is the radius.

Now, to find the rate of change of perimeter or, the circumference with respect to its radius, we need to differentiate the equation for circumference (C) that is (1), with respect to the radius (r). Thus,

$\frac{dC}{dr} =\frac{d}{dr} (2\pi r)$

$\implies \frac{dC}{dr} =2\pi \frac{d}{dr} (r)$

Now since

$\frac{d}{dr} (r) =1$

$\implies \frac{dC}{dr} =2\pi$

It can be seen that $\frac{dC}{dr}$ turns out to be constant that is $2\pi$, and, it will remain the same at all the values of r.

⇒ At r = 5 cm,

$\frac{dC}{dr} \Bigr|_{r=5} =2\pi$

Therefore, the rate of change of the perimeter of a circle with respect to its radius at r = 5 cm, is $2\pi .$