Multiple choice

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

  1. $1386 \: cm^{2}$
  2. $1388 \: cm^{2}$
  3. $1286 \: cm^{2}$
  4. $1186 \: cm^{2}$
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A Correct answer
Explanation

Area of equilateral triangle = (sqrt(3)/4) * s^2 = 484*sqrt(3). s^2 = 1936, s = 44. Perimeter = 3 * 44 = 132. Wire length = 132. For circle, 2 * pi * r = 132. r = 132 / (2 * 22/7) = 21. Area = pi * r^2 = 22/7 * 21 * 21 = 1386.

AI explanation

The area of an equilateral triangle is given by (sqrt(3)/4) * side^2, so setting this equal to 484*sqrt(3) gives a side length of 44 cm. The total length of the wire is the perimeter of the triangle, which is 3 * 44 = 132 cm, and this becomes the circumference of the circle (2 * pi * r = 132), yielding a radius of 21 cm. Finally, using the area formula for a circle, pi * r^2, we calculate the area as (22/7) * 21^2 = 1386 cm^2.

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