Question

A metal wire, when bent in the form of an equilateral triangle of largest area, encloses an area of $484\sqrt{3}c{m}^2$ . If the same wire is bent into the form of a circle of largest area, find the area of this circle.

• A. $1386c{m}^2$
• B. $1388c{m}^2$
• C. $1286c{m}^2$
• D. $1186c{m}^2$

Answer 1

The correct option is A $1386c{m}^2$

Let the side of the equilateral triangle be ${}^{\prime }{a}^{\prime }$.

Area of equilateral $△=\frac{\sqrt{3}}{4}{a}^2$

$\Rightarrow \frac{\sqrt{3}}{4}{a}^2=484\sqrt{3}\Rightarrow a^2=\frac{484\sqrt{3}\times 4}{\sqrt{3}}$

$\Rightarrow a^2=484\times 4\Rightarrow a=\sqrt{484\times 4}$

$\Rightarrow a=22\times 2=44cm$

Length of wire $=3a=3\times 44=132cm$

Also, length of wire $=$ circumference of the circle

$\Rightarrow 132=2\pi r$

$\Rightarrow r=\frac{66}{\pi }=\frac{66\times 7}{22}=21cm$

$\therefore$ Area of circle $=\pi r^2=\frac{22}{7}\times 21\times 21$

$=1386{cm}^2$