Question

Find the rate of change of the area of a circle with respect to its radius $r$ when $r = 3cm$ and $r = 4cm$.

Answer 1

Let the radius of the circle be r and the area of the circle be A.
We must calculate $\dfrac{{dA}}{{dr}}$ to determine the rate of change of Area concerning Radius.
Area of Circle $A = \pi {r^2}$ is a well-known formula.

Complete step-by-step solution:
Let r denote the radius of the circle and $A$ denotes the area of the circle.
The area of a circle ($A$) with radius ($r$) is calculated as follows:
$A = \pi {r^2}$
The area's rate of change concerning its radius is now given by,
$\dfrac{{dA}}{{dr}} = \dfrac{{d\left( {\pi {r^2}} \right)}}{{dr}}$ (we will go step by step derivation as follows)
$\dfrac{{dA}}{{dr}}\; = \;\pi \dfrac{{d\left( {{r^2}} \right)}}{{dr}}$ (taking out the constant term pie)
$\dfrac{{dA}}{{dr}} = \pi \left( {2r} \right)$ (differentiate concerning r)
$\dfrac{{dA}}{{dr}} = 2\pi r$
When $r = 3cm$
We have, $\dfrac{{dA}}{{dr}} = 2\pi r$
Putting $r = 3cm$
${\left. {\dfrac{{dA}}{{dr}}} \right|_{r\; = \;3}} = 2\pi \times 3$
${\left. {\dfrac{{dA}}{{dr}}} \right|_{r\; = \;3}} = 6\pi$
Since the area is measured in centimeters squared $({\text{c}}{{\text{m}}^2})$ and the radius is measured in centimeters $({\text{cm}})$,
${\left. {\dfrac{{dA}}{{dr}}} \right|_{r\; = \;3}} = 6\pi \;{\text{c}}{{\text{m}}^2}/cm$
As a result, when $r = 3cm$, Area increases at a rate of $6\pi \;{\text{c}}{{\text{m}}^2}/cm$.
When $r = 4cm$
We have, $\dfrac{{dA}}{{dr}} = 2\pi r$
Putting $r = 4cm$
${\left. {\dfrac{{dA}}{{dr}}} \right|_{r\; = \;4}} = 2\pi \times 4$
${\left. {\dfrac{{dA}}{{dr}}} \right|_{r\; = \;4}} = 8\pi$ (which is the required rate of change of the area of the circle)
Since the area is measured in centimeters squared $({\text{c}}{{\text{m}}^2})$ and the radius is measured in centimeters $({\text{cm}})$,
${\left. {\dfrac{{dA}}{{dr}}} \right|_{r\; = \;4}} = 8\pi \;{\text{c}}{{\text{m}}^2}/cm$
As a result, when $r = 4cm$, Area increases at a rate of $8\pi \;{\text{c}}{{\text{m}}^2}/cm$.
Hence, when the radius of the circle is $3cm$, the area of the circle changes at a rate of $6\pi \;{\text{c}}{{\text{m}}^2}/cm$ , and when the radius of the circle is $4cm$, the area of the circle changes at a rate of $8\pi \;{\text{c}}{{\text{m}}^2}/cm$.

Note: We must calculate $\dfrac{{dA}}{{dr}}$ to determine the rate of change of Area concerning Radius. Since the area is measured in centimeters squared $({\text{c}}{{\text{m}}^2})$. The radius is measured in centimeters $({\text{cm}})$, the unit for rate of change of area w.r.t radius of a circle is ${\text{c}}{{\text{m}}^2}/cm$.