Question 551049
<pre>
A goat is tied to one of the corners of a rectangular barn on a rope that is 50
feet long.  The dimensions of the barn are 40 feet by 30 feet.  Assuming that
the goat can graze wherever its rope allows it to reach, what is the square
footage of the grazing area for the goat?

{{{drawing(400,400,-55,55,-55,55,
green(line(0,40,0,50),line(0,0,0,-50), line(-30,0,-50,0),line(0,0,50,0)),

locate(-45,0,20ft),locate(-19,0,30ft),locate(20,0,50ft),locate(1,25,40ft),
locate(1,46,10ft), locate(1,-25,50ft), locate(-19,23,BARN),
line(-30,0,0,0),line(0,0,0,40),line(0,40,-30,40),line(-30,40,-30,0),
arc(0,0,100,-100,180,450), arc(-30,0,40,-40,90,180),
arc(0,40,20,-20,90,180) )}}}

The area of a circle is <font face = "symbol">p</font>r²

The grazing area consists of 

1. three quarters of a big 50ft-radius circle, which has area

    {{{3/4}}}·<font face = "symbol">p</font>(50)² = {{{3/4}}}·2500<font face = "symbol">p</font> square feet = 1875<font face = "symbol">p</font> square feet. 

2. one quarter of a 20ft-radius circle on the left, which has area

    {{{1/4}}}·<font face = "symbol">p</font>(20)² = {{{1/4}}}·400<font face = "symbol">p</font> square feet = 100<font face = "symbol">p</font> square feet 

3. one quarter of a small 10ft-radius circle on the top, which has area

    {{{1/4}}}·<font face = "symbol">p</font>(10)² = {{{1/4}}}·100<font face = "symbol">p</font> square feet = 25<font face = "symbol">p</font> square feet.  

Total = 1875<font face = "symbol">p</font> + 100<font face = "symbol">p</font> + 25<font face = "symbol">p</font> = 2000<font face = "symbol">p</font> square feet of grazing area.

Edwin McCravy aka AnlytcPhil</pre>