Question 217114
Step 1) Draw a 10' x 10' square to represent the shed. 


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/geo/step1.png"> 



Step 2) Extend a line segment from the midpoint of one side of the square (I chose the top) 18 ft out. This is the goat's leash. 


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/geo/step2.png"> 


Step 3) Now extend a segment 18 ft to the left (you could go either way). Take note how there's 5 feet from the midpoint to the corner. 


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/geo/step3.png"> 


Step 4) Sweep the two 18' segments (or just one) to form the segment shown in blue below. We'll call this area {{{A[1]}}}


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/geo/step4.png">
 


Step 5) Since the rope is eventually going to be caught on the shed, this means that we now have 13 feet (instead of 18') of rope. So just draw out an arbitrary 13' segment 


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/geo/step5.png"> 



Step 6) Sweep a circular segment through to get the red area shown below in which we'll call {{{A[2]}}}


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/geo/step6.png">
 


Step 7) Since 10 ft of rope is stuck on the wall, we now have only 3 ft of rope left. So sweep out the remaining area that the goat can roam (for this side at least). We'll call this new area in green  {{{A[3]}}}


<img src="http://i150.photobucket.com/albums/s91/jim_thompson5910/geo/step7.png">
 


Step 8) From the last drawing, we see that there are 3 quarter circles which have the radii 18, 13, and 3 feet. 


Recall that the area of a circle is {{{A=pi*r^2}}}. Divide this by 4 to find the area of a quarter circle (since all of these segments are quarter circles) to get {{{A=(pi*r^2)/4}}} 


Using the last formula, we then get the areas: 

 

{{{A[1]=(pi*(18)^2)/4=(pi*324)/4=81pi}}}



{{{A[2]=(pi*(13)^2)/4=(169pi)/4}}}



{{{A[3]=(pi*(3)^2)/4=(9pi)/4}}}
 
 


Now add the three individual areas: {{{A[1]+A[2]+A[3]=81pi+(169pi)/4+(9pi)/4=(251pi)/2}}}


So the area that the goat can graze on the entire left side is {{{(251pi)/2}}} square feet. 


To find the total area, simply multiply that last value by 2 (since the right side is a reflection of the left side) to get {{{2((251pi)/2)=(cross(2)*251pi)/cross(2)=251pi}}} 


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Answer: 

So the total area that the goat can graze is {{{251pi}}} square feet. If we use the approximation {{{3.14}}} for {{{pi}}} , then the approximate area is {{{251*3.14=788.14}}} square feet. 


So the total area available for the goat to graze is 788.14 square feet. 


Note: if you need better precision, then just use a better approximation of {{{pi}}}.