Very Difficult Are Question - please read the question carefully?

Question

A goat is grazing in a circular field with a rope attached to it of length r. The goat is free to eat all the grass as it moves around in the circular field and the area of the field is obviously pi.r^2.
Question - How long is another piece of the rope if it is now fixed to a point on the circumference of the circle and the goat can now only eat 50% of the grass in the original circle.

AREA question it should read.

Answer 1

Consider the upper half of the first circle

centered at origin O with radius 1 and let

point A , the center of the second circle ,

be at a distance of 1 to the right of O ; then let

point B be at an unknown but short estimated

distance to the left of O and set point C on the circle

such that all points on the curve from B to C

are equidistant from A and all points on the curve

from A to C are equidistant from O .

Finding angle BAC so that the sum of sector BAC and

the segment above line AC is equal to a quarter of the

original circle can be used to determine the radius of

the second circle as cosθ = r / 2r₁ , ( r₁ = 1 , r = 1+ x ) .

The area of the sector is ½ r ² θ = 2r₁²θcos²θ and

the area of the segment is

½ r ² ( π - θ ) - ½ r ² sin( π - θ ) = ½ r ² ( π - θ - sinθ ) .

The sum set equal to the area of the quarter circle is

2r₁²θcos²θ + ½ r ² ( π - θ - sinθ ) = ¼ π r ² and becomes

2θcos²θ + ¼ π - θ - sinθ = 0 .

Solving for θ in radians ( eg. Newton - Raphson )

and r = 2r₁cosθ ≈ 1.158 .

Answer 2

Hint:
Let L be the length of another piece of the rope and 2θ (rad) be the central angle that L intersects with the original circle.
(90-θ/2)L^2+[rθ^2-Lrcos(θ/2)] = 0.5 pi*r^2
where θ/2 = arcsin[L/(2r)].
Let r = 1. We can solve for L*.
L = L*r^2

Answer 3

okay, find the area for a length of rope that is, say, 10 feet long. After you find the area, divide by 2. Now work backwards and find the radius of this new area.

so 100pi
new area 50pi

answer=(square root of 50)/10

Answer 4

sq root of (r/2)

but i am not sure.. since its sayint 50% of the original circle...
so that means theres this 50% plus some other portion the goat can eat.. right?

Answer 5

S1=pi.r1^2
S2=pi.r2^2
S1=2S2
pi.r1^2=2pi.r2^2
r2=r1/(sqr(2))

Answer 6

square root of {(pi.r^2)/2}
not sure though