Question

A horse is tied to a vertex of a field having the shape of an equilateral triangle with a rope, the length of which is half of the length of a side of the triangle. The percentage of the total area of the field over which the horse can graze is approximately

• A. 20 to 22%
• B. 25 to 27%
• C. 30 to 32%
• D. 48 to 50%

Answer 1

Answer: (C)

30 to 32%

Let $D$ be the middle point of $AB$. Draw a circular arc $DE$. Then the horse can graze over the portion $ADEA$.
Let $AB=x$. Then $AD=\frac{x}{2}$.
$\therefore$ Area of portion $ADEA=\frac{\pi }{6}.(\frac{x}{2}{)}^2=\frac{\pi x^2}{24}$ and total area of the field $=\frac{x^2\sqrt{3}}{4}$
Required % $=\frac{24}{\frac{x^2\sqrt{3}}{4}}\times 100$
$=\frac{\pi }{6\sqrt{3}}\times 100$
$=30.2$ % (approx.)