Question

The circumference of a circle is directly proportional to the length of its radius, while the area it encloses is directly proportional to the square of the length of its radius. If the area of a circle is multiplied by $49$, then its circumference is multiplied by ______.

Answer 1

Step-1: Find the radius:

The area $A$ encloses is directly proportional to the square of the length of its radius $r$.

$A\propto r^2$

$\Rightarrow$ $\sqrt{A}\propto r\left(\mathrm{Taking}\mathrm{square}\mathrm{root}\mathrm{of}\mathrm{both}\mathrm{sides}\right)$

$\therefore$ $r\propto \sqrt{A}$

Also the area of a circle is multiplied by $49$.

Therefore, multiply $49$ by $A$ in $r\propto \sqrt{A}$ and simplify.

$r\text{'}\propto \sqrt{A\times 49}\left[r\text{'}\mathrm{is}\mathrm{the}\mathrm{new}\mathrm{radius}\right]$

$\Rightarrow$$r\text{'}\propto 7\sqrt{A}\left[\because \sqrt{49}=7\right]$

$\therefore$$r\text{'}\propto 7r$

Step-2: Find the Circumference:

Since circumference $C$ of a circle is directly proportional to the length of its radius, $C\propto r$,

where $k$ is proportionality constant, and $r$ is the radius of a circle.

As the new radius, $r\text{'}$ is proportional to $7r$. Therefore, multiply both sides by $7$ in $C\propto r$ and simplify.

$C\propto r$

$\Rightarrow$$7C\propto 7r$

$\therefore$ $C\text{'}\propto 7r\left[C\text{'}=7C\right]$

Hence, if the area of a circle is multiplied by $49$, then its circumference is multiplied by $7$.