Question

a goat tethered to a stake by a rope has grazed the meadow over an area of 12.56msqare around it stretching the rope. i) what is the length of the rope? ii)what is the perimeter of the meadow?

Answer 1

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

Answer 2

Answer:

2m

12.56 m

Step-by-step explanation:

A= 12.56 m²

r=? P=?

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A= πr²

1. 12.56=3.14r² ⇒ r²=12.56/3.14 ⇒ r²=4 ⇒ r=2 m
2. P= 2πr= 2*3.14*2m= 12.56 m