Answers:
length of rope = 2 meters
perimeter of meadow = 12.56 square meters
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Explanation:
Assuming there are no structures (buildings, fences, etc) to get in the way of the goat, this means the goat's area that it can roam is a circle with radius r
The area of the circle is pi*r^2 which is equal to 12.56 square meters
pi*r^2 = 12.56
3.14*r^2 = 12.56
r^2 = 4
r = sqrt(4)
r = 2
The radius of the circle is 2 meters. This is the length of the rope. If the goat goes out as far as it can go, and tries to move left/right, then it will trace out a circle of radius 2. This is assuming the rope is kept fully taut the entire time.
The perimeter of the circle is the same as the circumference
circumference = 2*pi*r
circumference = 2*pi*2
circumference = 4*pi
circumference = 4*3.14
circumference = 12.56
The circumference and area are the same because r^2 = 4 and 2*r = 4 when r = 2. For any other positive r value, the area and circumference will be different values.
So in short, if a friend says that the area of a circle is the same as its perimeter, then you can immediately conclude the radius is 2.
Side note: throughout all of this solution, I used the approximation pi = 3.14 though pi has more decimal digits
