Question 616393
zoe the goat is tied to a rope to one corner of a 15 meter by 25 meter rectangular barn. over what area of the field can zoe graze is the rope is
1. 10 meters long?
2. 20 meters long?
3. 30 meters long?
4. zoe is happiest when she has at least 400 meters squared to graze. what possible lengths of rope could be used?

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1. 10 meters long?
<pre>
{{{drawing(400,400,-35,30,-35,30,

rectangle(0,0,25,15), arc(0,0,20,-20,90,360)


 )}}}

That's 3/4ths of a circle with a radius of 10.

The area of a full circle with radius 10 is

A = {{{pi*r^2}}} = {{{pi*10^2}}} = {{{100pi}}} 

Multiply that by 3/4ths gives {{{3/4}}}·{{{100pi}}} = {{{75pi}}} = 235.6 mē 

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2. 20 meters long?
<pre>
{{{drawing(400,400,-35,30,-35,30, green(line(0,15,0,20)),

rectangle(0,0,25,15), arc(0,0,40,-40,90,360),
arc(0,15,10,-10,0,90)


 )}}}

That's 3/4ths of a circle with a radius of 20. plus 1/4th of a circle
with a radius of 5

The area of a full circle with radius 20 is

A = {{{pi*r^2}}} = {{{pi*20^2}}} = {{{400pi}}} 

Multiply that by 3/4ths gives {{{3/4}}}·{{{400pi}}} = {{{300pi}}} = 942.48 mē 

The area of a full circle with radius 5 is

A = {{{pi*r^2}}} = {{{pi*5^2}}} = {{{25pi}}} 

Multiply that by 1/4th gives {{{1/4}}}·{{{25pi}}} = {{{6.25pi}}} = 19.63 mē

Total area = 942.48+19.63 = 962.11 mē.

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3. 30 meters long?
<pre>
{{{drawing(400,400,-35,30,-35,30, green(line(0,15,0,30)),

rectangle(0,0,25,15), arc(0,0,60,-60,90,360),
arc(0,15,30,-30,0,90), arc(25,0,10,-10,0,90), green(line(25,0,30,0))


 )}}}

That's 3/4ths of a circle with a radius of 30.