geometry
A goat is tied to the corner of a small house, with a 6 m long rope . The house is 3 m wide by 4 m long its rektanguler house. There is grass around the house.
On how much property can the goat graze?
I got the answer 95m2 but that answer is wrong so i dont really know anymore
The area of the half circle should be no problem. The shape that is made when the cow wraps around is called the involute of a circle. Using some trig/geometry you can parametrize the curve as $x=r\cos t+tr\sin t$ and $y=r\sin t-tr\cos t$. Then use Green's theorem for area $A=\frac{1}{2}\int_C-ydx+xdx.$ This is actually a great problem and can be solved a couple of other ways as well.
If you want to use Riemann sums you will be summing up the area of circular sectors as the angle gets smaller and smaller. Here is a crude picture when
$n=4$ and $r$ is the radius of the barn. Taking the limit of the sum as $n\rightarrow\infty$ will get you the answer.
Parts of three different circles.
Best Answer
As everyone suggests, once you draw a picture, the answer will be obvious.
The area the goat can graze will split into 3 circular sectors,
This means the total area is
$$\frac34\pi (6)^2 + \frac14\pi(3)^2 + \frac14\pi(2)^2 = \frac{121}{4}\pi \approx 95.03317777109125 \text{(in sq. meter)}$$