SOLUTION: 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

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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?
Found 2 solutions by EdwinParker, Edwin McCravy:
Answer by EdwinParker(16) About Me  (Show Source):
You can put this solution on YOUR website!
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?

1. 10 meters long?


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

The area of a full circle with radius 10 is

A = pi%2Ar%5E2 = pi%2A10%5E2 = 100pi 

Multiply that by 3/4ths gives 3%2F4·100pi = 75pi = 235.6 mē 

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2. 20 meters long?


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%2Ar%5E2 = pi%2A20%5E2 = 400pi 

Multiply that by 3/4ths gives 3%2F4·400pi = 300pi = 942.48 mē 

The area of a full circle with radius 5 is

A = pi%2Ar%5E2 = pi%2A5%5E2 = 25pi 

Multiply that by 1/4th gives 1%2F4·25pi = 6.25pi = 19.63 mē

Total area = 942.48+19.63 = 962.11 mē.

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3. 30 meters long?






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

Answer by Edwin McCravy(20055) About Me  (Show Source):
You can put this solution on YOUR website!
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?

1. 10 meters long?


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

The area of a full circle with radius 10 is

A = pi%2Ar%5E2 = pi%2A10%5E2 = 100pi 

Multiply that by 3/4ths gives 3%2F4·100pi = 75pi = 235.6 mē 

---------------------------------------------

2. 20 meters long?


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%2Ar%5E2 = pi%2A20%5E2 = 400pi 

Multiply that by 3/4ths gives 3%2F4·400pi = 300pi = 942.48 mē 

The area of a full circle with radius 5 is

A = pi%2Ar%5E2 = pi%2A5%5E2 = 25pi 

Multiply that by 1/4th gives 1%2F4·25pi = 6.25pi = 19.63 mē

Total area = 942.48+19.63 = 962.11 mē.

-----------------------------------------------------------

3. 30 meters long?


That's 3/4ths of a circle with a radius of 30. plus 1/4th of a circle
with a radius of 15 at the top, plus 1/4 of a circle with radius 5 on
the right.

The area of a full circle with radius 30 is

A = pi%2Ar%5E2 = pi%2A30%5E2 = 900pi 

Multiply that by 3/4ths gives 3%2F4·900pi = 900pi = 2827.43 mē 

The area of a full circle with radius 10 is

A = pi%2Ar%5E2 = pi%2A10%5E2 = 100pi 

Multiply that by 1/4th gives 1%2F4·100pi = 25pi = 78.54 mē

The area of a full circle with radius 5 is

A = pi%2Ar%5E2 = pi%2A5%5E2 = 25pi 

Multiply that by 1/4th gives 1%2F4·25pi = 6.25pi = 19.63 mē

Total area = 2827.43 mē + 78.54 mē + 19.63 mē = 2925.60 mē.

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zoe is happiest when she has at least 400 meters squared to graze. what possible lengths of rope could be used?
To have exactly 400 mē of grazing area, it would be like the first case,
where we just have three-quarters of just one circle.  Let the radius
be R.  

The area of a full circle with radius R is

A = pi%2AR%5E2  

Multiply that by 3/4th gives 3%2F4·pi%2AR%5E2 

Setting that = 400

3%2F4·pi%2AR%5E2 = 400

Multiply through by 4

3pi%2AR%5E2 = 1600

       Rē = 1600%2F%283pi%29
       Rē = 169.7652726
        R = 13.03 m

So the rope would have to be 13.03 m long or longer.

Edwin