A goat tied to acircular plot of dia 20m it grazes 1/2 of plot.Find length of rope.?

Answer 1

First the area of the plot is given by the following formula, and a radius of 10:
A = pi R²
A = pi 10²
A = 100 pi
So the original circular plot has an area of 100 pi sq. meters. It's not necessary to figure out the decimal equivalent, because pi will cancel out in the step below.

So now we need to find a circle with half that area. In other words, what value of r will make a circle with an area of 50 pi.

Let r = length of rope.
A = pi r²
50 pi = pi r²

Cancel out the pi
50 = r²

Take the square root of both sides (but since a length is always positive, you can ignore the negative.

r = sqrt(50)
r = 5*sqrt(2)
r = 7.07106781 m

A rope with a length of 7.07m will allow the goat to graze half the original plot.

Answer 2

there is not actually enough information here to resolve this

it is different depending on where the goat is tied

in the middle of the plot or at the circumference

if you assume goat is tied to a stake near the middle of the plot and thus the grazed area is a circle fully inside the plot then there is a single answer

the area of the circular plot is:
pi*r^2
3.14*10^2=314 m^2

the area that the goat eats is half that, or 157 m^2

for a circle with area 157 m^2:

pi*r^2=157
3.14*r^2=157
r^2=157/3.14=50
r=sqrt(50)=7.07 m

the rope length would be the radius of the grazed circle so:
7.07 m is the answer

you can do this a little more elegantly by solving the algebra first and then sticking in the numbers, but I think this way is a little clearer

Answer 3

kesav, I think you forgot to mention that the stake end of the rope is located on the edge of the circular plot. This makes it a most interesting problem that I ran across a few years ago.

It can be solved without calculus (good thing; the integrals get REALLY messy!), but it is still quite a challenge. I believe there's no closed form solution; you have to iterate a trig equation to get a closer & closer solution. The answer comes out about 11.587 m for your field radius of 10 m.

Sorry--don't have the cyber-savy to post a diagram on the web for all to look at, but can do so through a direct contact.

Answer 4

the radius of the circular plot will be 10 m
and area will be 22/7*10*10=3.14*10*10=314 square metres
it gazed half

therefore the area it grazed will be 314/2 = 157

157 = 3.14 * (rope )^2

(rope)^2 = 157/3.14 =50

therefore length of rope = (50)^1/2 = 7.071 metres

Answer 5

Making the assumption that it is tied to the centre of the plot (that isn't specified);
Area of a circle = pi x radius x radius. So this circle's area is pi x 10 x 10, or approx 314 sq m. (I haven't got a calculator here).
So you need to find the radius of a circle of half that area, or about 157 sq m.
Just work out the sum, pi x radius x radius = 157. In my head that will be something like 7 m.

Answer 6

Area of the plot = (22/7)*20^2 =>1257.14m^2
1/2 Area = 628.5m^2
Length of the rope = SQRT(628.5/(22/7))=>14.14m

Answer 7

it extremely is nearly a million/4 the element of a circle. i'm assuming it could in basic terms graze contained in the sq.. The radius of the circle is the 7 meter rope. element of a circle pi r^2 = pi 40 9 = 483.595 sq. meters. section /4 = one hundred twenty.898 sq. meters. The 12 meters says it extremely is contained in the sq..

Answer 8

The goat would munch the rope through so it could graze wherever it wanted. ;-)

Answer 9

Well its simple,
Dia of plot = 20,
Therefore area = 400*3.14
therefore half area = 200*3.14

Therefore radius = sqrt (200*3.14/3.14)
= 7.071

Answer 10

the first answer is correct