Question

A HORSE IS TIED TO ONE CORNER OF THE CIRCULAR BARN OF RADIUS 8M WITH THE ROPE OF 12 M FIND THE AREA WHICH THE HORSE CAN GRAZE INSIDE THE BARN.

Answer 1

Given : A HORSE IS TIED TO ONE CORNER OF THE CIRCULAR BARN OF RADIUS 8M WITH THE ROPE OF 12 M

To find : THE AREA WHICH THE HORSE CAN GRAZE INSIDE THE BARN.

Solution:

Radius of one circle = 8   m

Radius of horse rope = 12 m

Angle formed at center α

12² = 8² + 8² - 2*8*8Cosα

=> 144 = 64 + 64 - 128Cosα

=> Cosα = -1/8

=> α =  Cos⁻¹(1/8) = 97.18°

Area by two sectors = 2 * ( 97.18°  / 360° ) *π * 8²

= 108.58  m²  

Third angle at center = 360° - 2( 97.18° )  

angle by chord at center = 2 * angle by chord at arc segment

Angle formed at point where horse is tied   =  (1/2)(360° - 2( 97.18° )  )

= (180° - 97.18° ) = 82.82°

Area = (  82.82° / 360°) π * 12²  = 104.07 m²

Area of two triangles to be reduced of Sides 8 , 8 & 12 m

= 31.75

2 * 31.75  = 63.5 m²

Area Horse can graze  = 108.58  + 104.07   - 63.5

= 149.15 m²  

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Answer 2

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The area available for horse to graze is nothing but "Area of Quadrant of a circle'

∴ Area of Quadrant = π×r

3.14×5×5

=19.625m

If the length of rope is increased to 10m then the new radius ,=10m

∴ Area of new quadrant = 4

3.14×10×10

=78.5m

∴ Increase in grazing area =78.5−19.625=58.875