question_answer

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%

Correct Answer: C

Solution :

 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 \[ADEA=\frac{\pi }{6}.{{\left( \frac{x}{2} \right)}^{2}}=\frac{\pi {{x}^{2}}}{24}\] and total area of the field \[=\frac{{{x}^{2}}\sqrt{3}}{4}\] Required \[%=\frac{\frac{\pi {{x}^{2}}}{24}}{\frac{{{x}^{2}}\sqrt{3}}{4}}\times 100\]                 \[=\frac{\pi }{6\sqrt{3}}\times 100\] \[=30.2%\] (approx.)