Question

The area of a circle is given by A(r)=pie^2 and the radius in terms of the circumference is given by r(C)=C/2pie. Find A(r(C))

Answer 1

`\large\boxed{A(r(C))=(C^2)/(4\pi)}`

Explanation:

`A(r)=\pi r^2\\\\r(C)=(C)/(2\pi)\\\\A(r(C))\to\text{put}\ r=(C)/(2\pi)\ \text{to}\ A(r):\\\\A(r(C))=\pi\left((C)/(2\pi)\right)^2\qquad\text{use}\ \left((a)/(b)\right)^n=(a^n)/(b^n)\ \text{and}\ (ab)^n=a^nb^n\\\\A(r(C))=\pi\cdot(C^2)/(2^2\pi^2)\qquad\text{cancel one}\ \pi\\\\A(r(C))=(C^2)/(4\pi)`